notes electrostatics.pdf

# Fundamental concepts of electrostatics Electrostatics deals with the study of stationary electric charges and the forces that arise between them. This section introduces the fundamental principles governing these interactions, including the nature of electric charge, Coulomb's law describing the force between charges, and the concept of the electric field [1](#page=1). ### 1.1 Electric charge Electric charge is a fundamental property of matter that causes it to experience a force when placed in an electric or magnetic field. Charges can be positive or negative. The SI unit of electric charge is the coulomb (C). One coulomb is equivalent to the charge of approximately $6.25 \times 10^{18}$ electrons [1](#page=1). ### 1.2 Coulomb's law Coulomb's law quantifies the force between two point charges. It states that the magnitude of the electrostatic force between two point charges is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them. The force acts along the line connecting the two charges [1](#page=1). The formula for Coulomb's law is: $$F = k \frac{|q_1 q_2|}{r^2}$$ where: * $F$ is the magnitude of the electrostatic force in Newtons (N). * $k$ is Coulomb's constant, approximately $9 \times 10^9 \, \text{N m}^2/\text{C}^2$ [1](#page=1). * $q_1$ and $q_2$ are the magnitudes of the two charges in coulombs (C). * $r$ is the distance between the centers of the two charges in meters (m). Coulomb's constant $k$ can also be expressed in terms of the permittivity of free space, $\epsilon_0$: $$k = \frac{1}{4 \pi \epsilon_0}$$ The value of $\epsilon_0$ is approximately $8.852 \times 10^{-12} \, \text{C}^2/\text{N m}^2$ [1](#page=1). #### 1.2.1 Dielectric constant The dielectric constant ($K$ or $\epsilon_r$) of a medium is a dimensionless quantity that describes how an electric field affects and is affected by that medium. It is defined as the ratio of the permittivity of the medium ($\epsilon$) to the permittivity of free space ($\epsilon_0$): $$K = \epsilon_r = \frac{\epsilon}{\epsilon_0}$$ This implies that the permittivity of the medium is $\epsilon = K \epsilon_0$ [1](#page=1). The presence of a dielectric medium reduces the force between charges compared to that in a vacuum. If $F_{vacuum}$ is the force in a vacuum and $F_{medium}$ is the force in a medium with dielectric constant $K$, then: $$F_{medium} = \frac{F_{vacuum}}{K}$$ Some common values for dielectric constants include: * Air: $K_{air} = 1$ (approximately, so $F_{vacuum} \approx F_{air}$) [1](#page=1). * Water: $K_{water} = 81$ [1](#page=1). * Conductor: $K_{conductor} = \infty$ (implying $F_{metal} = 0$ for conductors, as the electric field inside is zero) [1](#page=1). The interaction distance ($d$) in a medium can be related to the vacuum distance ($r$) by $d = r\sqrt{K}$ [1](#page=1). > **Tip:** The dielectric constant is a crucial factor when considering electrostatic forces in different materials. A higher dielectric constant signifies a greater ability of the material to reduce the electric field strength. ### 1.3 Electric field The electric field ($\vec{E}$) at a point in space is defined as the force per unit positive test charge experienced at that point. It is a vector quantity, meaning it has both magnitude and direction. The direction of the electric field is the direction of the force that would be exerted on a positive test charge placed at that point [1](#page=1). The electric field can be calculated using the formula: $$\vec{E} = \frac{\vec{F}}{q_0}$$ where $\vec{F}$ is the electrostatic force on a test charge $q_0$ [1](#page=1). The electric field is formally defined as the limit of the force per unit charge as the test charge approaches zero: $$\vec{E} = \lim_{q_0 \to 0} \frac{\vec{F}}{q_0}$$ For a point charge $q$, the magnitude of the electric field at a distance $r$ is given by: $$E = k \frac{|q|}{r^2}$$ The direction of $\vec{E}$ is radially outward from a positive charge and radially inward towards a negative charge [1](#page=1). #### 1.3.1 Electric field due to multiple charges The electric field due to a system of point charges is the vector sum of the electric fields produced by each individual charge at that point (superposition principle) [1](#page=1). $$\vec{E}_{total} = \vec{E}_1 + \vec{E}_2 + \dots + \vec{E}_n$$ #### 1.3.2 Null point A null point is a location in space where the net electric field due to all the charges in a system is zero. For two like charges ($q_1$ and $q_2$, both positive or both negative), the null point lies on the line connecting the charges, between them. Its distance ($z$) from $q_1$ can be found using the formula [1](#page=1): $$z = \frac{r}{1 + \sqrt{\frac{q_2}{q_1}}}$$ where $r$ is the distance between $q_1$ and $q_2$. For two unlike charges, the null point lies on the line extending from the charges, beyond the smaller charge [1](#page=1). ### 1.4 Charged particles in uniform electric fields When a charged particle is placed in a uniform electric field, it experiences a constant force, leading to acceleration. #### 1.4.1 Acceleration and velocity The acceleration ($a$) of a charged particle of mass $m$ and charge $q$ in a uniform electric field $E$ is given by: $$a = \frac{qE}{m}$$ The velocity ($v$) of the particle at time $t$, starting from rest, is: $$v = at = \frac{qEt}{m}$$ The displacement ($s$) at time $t$ is: $$s = \frac{1}{2}at^2 = \frac{1}{2}\frac{qEt^2}{m}$$ The relationship between velocity, acceleration, and displacement is: $$v^2 = 2as = 2 \left(\frac{qE}{m}\right) s$$ #### 1.4.2 Work done and power The work done ($W$) on a charged particle moving a displacement $s$ in a uniform electric field $E$ is: $$W = \vec{F} \cdot \vec{s} = q \vec{E} \cdot \vec{s}$$ If the force and displacement are in the same direction, $W = qEs$. This work done is also related to the change in kinetic energy: $$W = \Delta KE = \frac{1}{2} m v^2$$ So, $W = \frac{1}{2} m (\frac{qEt}{m})^2 = \frac{q^2 E^2 t^2}{2m}$ [1](#page=1). The instantaneous power ($P$) delivered by the electric field is: $$P = \vec{F} \cdot \vec{v} = q \vec{E} \cdot \vec{v}$$ If $\vec{E}$ and $\vec{v}$ are in the same direction, $P = qEv$ [1](#page=1). #### 1.4.3 Charged particle entering a uniform electric field When a charged particle enters a uniform electric field, its motion depends on its initial velocity and charge. * **Parallel entry (supporting or retarding force):** If the initial velocity is parallel to the field, the particle accelerates if the force supports the motion ($qE/m > 0$) or decelerates if the force opposes it ($qE/m < 0$) [2](#page=2). * **Supporting force (+ve charge):** $a = \frac{qE}{m} > 0$. $v = u + at$. $W = +ve$ [2](#page=2). * **Retarding force (-ve charge):** $a = -\frac{qE}{m} < 0$. $v = u - \frac{qE}{m}t$. $W = -ve$. If $v=0$ (momentary rest), $v = -u$ can occur if $u$ is directed against the field [2](#page=2). * **Perpendicular entry:** If the particle enters the field perpendicular to the field lines with initial velocity $u$, its motion can be parabolic or result in linear and angular deflection. * **Parabolic path:** The particle undergoes uniform acceleration perpendicular to its initial velocity. The trajectory is parabolic [2](#page=2). * Velocity components: $v_x = u$, $v_y = \frac{qE}{m}t$. * Position components: $x = ut$, $y = \frac{1}{2} \frac{qE}{m} t^2 = \frac{qEx^2}{2mu^2}$. * The height of deflection ($h$) is given by $h = \frac{2EQ^2}{2mu^2}$ [2](#page=2). * **Linear deflection:** If the particle exits the field after traveling a distance $L$ parallel to the field, the linear deflection is given by $h = \frac{qEL^2}{2mu^2}$ [2](#page=2). * **Angular deflection:** The angle of deflection ($\theta$) can be found using $\tan \theta = \frac{v_y}{v_x} = \frac{qEL}{mu^2}$ [2](#page=2). ### 1.5 Electric dipole An electric dipole consists of two equal and opposite point charges, $+q$ and $-q$, separated by a small distance $2l$. The electric dipole moment ($\vec{p}$) is a vector quantity defined as the product of the magnitude of one charge and the distance between the charges. Its direction is conventionally from the negative charge to the positive charge [2](#page=2). $$p = q(2l)$$ The magnitude of the dipole moment is $p = q(2l)$ [2](#page=2). #### 1.5.1 Electric field due to a dipole The electric field produced by a dipole varies with position. * **Axial point (on the axis of the dipole):** The electric field is directed along the axis and its magnitude is: $$E_{axial} = \frac{2kpr}{ (r^2 - l^2)^2 } \approx \frac{2kp}{r^3}$$ for a short dipole ($r \gg l$) [2](#page=2). * **Equatorial point (on the perpendicular bisector):** The electric field is opposite to the dipole moment and its magnitude is: $$E_{equatorial} = \frac{kp}{(r^2 + l^2)^{3/2}} \approx \frac{kp}{r^3}$$ for a short dipole ($r \gg l$) [2](#page=2). * **At any point:** The electric field magnitude is $E = \frac{kp}{r^3} \sqrt{1 + 3\cos^2\theta}$, where $\theta$ is the angle between the dipole moment vector and the position vector. The direction can be found by $\tan \phi = \frac{1}{2}\tan\theta$ [2](#page=2). #### 1.5.2 Dipole in a uniform electric field When an electric dipole is placed in a uniform external electric field ($\vec{E}_{ext}$), it experiences a net torque ($\vec{\tau}$) that tends to align it with the field. The torque is given by: $$\vec{\tau} = \vec{p} \times \vec{E}_{ext}$$ The magnitude of the torque is $\tau = p E_{ext} \sin\theta$, where $\theta$ is the angle between $\vec{p}$ and $\vec{E}_{ext}$ [2](#page=2). * If $\theta = 0^\circ$ or $180^\circ$, the net torque is zero ($\tau = 0$). * If $\theta = 90^\circ$, the torque is maximum ($\tau = PE_{max}$) [2](#page=2). #### 1.5.3 Work done to rotate a dipole Work must be done to rotate a dipole in a uniform electric field against the aligning torque. The work done ($W$) in rotating the dipole from an initial angle $\theta_1$ to a final angle $\theta_2$ is: $$W = \int_{\theta_1}^{\theta_2} \tau d\theta = pE_{ext}(\cos\theta_1 - \cos\theta_2)$$ This work done is equal to the change in potential energy: $W = -\Delta U = U_{final} - U_{initial}$ [2](#page=2). #### 1.5.4 Potential energy of a dipole The absolute potential energy ($U$) of an electric dipole in a uniform electric field is given by: $$U = -\vec{p} \cdot \vec{E}_{ext} = -pE_{ext}\cos\theta$$ [2](#page=2). * **Stable equilibrium:** When $\theta = 0^\circ$, $U = -pE_{ext}$ (minimum potential energy) [2](#page=2). * **Unstable equilibrium:** When $\theta = 180^\circ$, $U = +pE_{ext}$ (maximum potential energy) [2](#page=2). #### 1.5.5 Force on a dipole in a non-uniform field In a uniform electric field, the net force on a dipole is zero. However, in a non-uniform electric field, there is a net force. The force on a dipole in a non-uniform field is approximately: $$F_{net} \approx \pm \frac{dp}{dr} E(r)$$ or more precisely, $F_{net} = \frac{dp}{dr}$ where $p$ is the component of dipole moment along the field gradient [2](#page=2). --- # Electric fields due to charge distributions This topic explores the calculation of electric fields arising from various continuous charge arrangements, moving beyond point charges to address more complex charge configurations [3](#page=3). ### 2.1 Continuous charge distributions For continuous charge distributions, the total charge is not a discrete sum but an integral over the charge density. Charge density can be defined in three ways [3](#page=3): * **Linear charge density ($\lambda$)**: Charge per unit length. It is defined as $\lambda = \frac{dQ}{dl}$ [3](#page=3). * **Surface charge density ($\sigma$)**: Charge per unit area. It is defined as $\sigma = \frac{dQ}{dA}$ [3](#page=3). * **Volume charge density ($\rho$)**: Charge per unit volume. It is defined as $\rho = \frac{dQ}{dV}$ [3](#page=3). ### 2.2 Calculating electric fields from charge distributions To find the electric field ($\vec{E}$) due to a continuous charge distribution, we consider an infinitesimal charge element ($dq$) and calculate the electric field it produces at a point. This infinitesimal field ($d\vec{E}$) is then integrated over the entire charge distribution. The general form is $d\vec{E} = k \frac{dq}{r^2} \hat{r}$, where $r$ is the distance from the charge element to the point of interest, and $\hat{r}$ is the unit vector pointing from the charge element to the point [3](#page=3). #### 2.2.1 Field due to a line of charge For a finite line of charge, the electric field components can be calculated using integration. Consider a point on the perpendicular bisector of the line segment [3](#page=3). * **General formula for components (at a distance $y$ from midpoint):** Let the line charge extend from angle $-\theta_1$ to $\theta_2$ with respect to the point of observation. * $E_x = k\lambda \int \frac{\cos\theta \, dl}{r^2}$ * $E_y = k\lambda \int \frac{\sin\theta \, dl}{r^2}$ A more practical form for components along the perpendicular bisector ($x$-axis) and parallel to the line ($y$-axis) at a distance $y$ from the midpoint of a line segment of length $L$ at $x=0$ is: $E_x = k\lambda \left( \sin\theta_1 + \sin\theta_2 \right)$ [3](#page=3). $E_y = k\lambda \left( \cos\theta_1 - \cos\theta_2 \right)$ [3](#page=3). * **Specific Cases for a line charge:** * **Point on the perpendicular bisector of a finite line charge:** If the line has length $2a$ and the point is at a distance $y$ from the center, $\theta_1 = \theta_2 = \theta$. $E_x = 2k\lambda \frac{\sin\theta}{y}$ where $\sin\theta = \frac{a}{\sqrt{y^2 + a^2}}$. Thus, $E_x = \frac{2k\lambda a}{y\sqrt{y^2+a^2}}$. $E_y = 0$ [3](#page=3). * **Infinite line of charge:** Here, $\theta_1 = \pi/2$ and $\theta_2 = -\pi/2$. $E_x = k\lambda (\sin(\pi/2) + \sin(-\pi/2)) = k\lambda (1 - 1) = 0$. $E_y = k\lambda (\cos(\pi/2) - \cos(-\pi/2)) = k\lambda (0 - 0) = 0$. This formulation seems incorrect in the document. Using $r = y/\cos\theta$ and $dl = y d\theta/\cos^2\theta$, $dq = \lambda dl = \lambda y d\theta / \cos^2\theta$. $dE_x = dE \cos\theta = k \frac{\lambda y d\theta}{\cos^2\theta} \frac{\cos\theta}{(y/\cos\theta)^2} = k \lambda \frac{\cos\theta d\theta}{y}$. Integrating from $-\pi/2$ to $\pi/2$ for an infinite line yields $E_x = \frac{k\lambda}{y} [\sin\theta]_{-\pi/2}^{\pi/2} = \frac{2k\lambda}{y}$ [3](#page=3). Using Gauss's law for an infinite line of charge, the electric field is found to be $E = \frac{2k\lambda}{r}$ directed radially outwards for a positive charge [3](#page=3). * **Point at the end of a long line charge:** If the point is at the end of a long line, one angle is 0 and the other approaches $\pi/2$. $E_x = k\lambda \sin(\pi/2) = k\lambda$ $E_y = k\lambda (1 - \cos(\pi/2)) = k\lambda$ This is likely for a semi-infinite line. The document states $E_x = k\lambda / r$ and $E_y = k\lambda / r$ for a point at the end of a long line, which is simplified [3](#page=3). #### 2.2.2 Field due to a charged ring For a charged ring of radius $R$ and total charge $Q$, the electric field along the axis passing through the center and perpendicular to the plane of the ring at a distance $x$ from the center is: $$ E_x = \frac{kQx}{(R^2 + x^2)^{3/2}} $$ The electric field components perpendicular to the axis are zero due to symmetry ($E_y = E_z = 0$) [3](#page=3). * At the center of the ring ($x=0$), $E_x = 0$ [3](#page=3). * The maximum electric field occurs at $x = R/\sqrt{2}$ [3](#page=3). #### 2.2.3 Field due to a charged arc For a uniformly charged arc subtending an angle $2\phi$ at the center of radius $R$, the electric field at the center is: $$ E = 2k\lambda \frac{\sin(\phi/2)}{R} $$ where $\lambda$ is the linear charge density. * **Semicircular ring:** For a semicircle, $2\phi = \pi$, so $\phi = \pi/2$. $E = 2k\lambda \sin(\pi/4) = 2k\lambda (\frac{1}{\sqrt{2}}) = \sqrt{2} k\lambda$ [3](#page=3). * **Quadrant ring:** For a quadrant, $2\phi = \pi/2$, so $\phi = \pi/4$. $E = 2k\lambda \sin(\pi/8)$. The document states $E = \sqrt{2} k \lambda / R$, which appears to be a simplified or specific case for a quadrant [3](#page=3). #### 2.2.4 Field due to a charged disk For a uniformly charged disk of radius $R$ and surface charge density $\sigma$, the electric field at a point on the axis perpendicular to the disk at a distance $x$ from the center is: $$ E = \frac{k\sigma x}{2} \left( \frac{1}{\sqrt{R^2 + x^2}} - \frac{1}{x} \right) $$ This formula seems to have a typo. The correct derivation using integration of infinitesimally thin rings gives: $$ E_x = \frac{k\sigma x}{2} \left( \frac{1}{\sqrt{R^2 + x^2}} \right) $$ The document provides $E = \frac{k x}{\sqrt{R^2+x^2}}$, which appears to be missing the $\sigma/2$ factor [3](#page=3). * **At the center of the disk ($x=0$):** $E = \frac{k\sigma}{2}$ [3](#page=3). * **For a very large disk (approaching an infinite sheet):** As $R \to \infty$, $\frac{1}{\sqrt{R^2+x^2}} \to 0$ and $\frac{1}{x}$ dominates, leading to an infinite field if the formula is used directly. However, if the original form $E_x = \frac{k\sigma x}{2} (\frac{1}{\sqrt{R^2+x^2}})$ is considered, then as $R \to \infty$, $\sqrt{R^2+x^2} \approx R$, and $E_x \to \frac{k\sigma x}{2R}$, which tends to zero. The electric field of an infinite sheet is $\frac{\sigma}{2\epsilon_0} = \frac{k\sigma}{2} \times 2\epsilon_0$. The document formula for the disk at the center is $E = \sigma / 2\epsilon_0$, which is the field of an infinite sheet [3](#page=3). ### 2.3 Electric flux and Gauss's theorem * **Electric flux ($\Phi$)**: A measure of the electric field passing through a given area. It is defined as the dot product of the electric field and the area vector: $\Phi = \vec{E} \cdot \vec{A}$ [3](#page=3). * **Gauss's Theorem**: States that the total electric flux through any closed surface (Gaussian surface) is proportional to the total electric charge enclosed within that surface. Mathematically: $$ \oint \vec{E} \cdot d\vec{A} = \frac{Q_{enc}}{\epsilon_0} $$ or using Coulomb's constant $k = \frac{1}{4\pi\epsilon_0}$: $$ \oint \vec{E} \cdot d\vec{A} = 4\pi k Q_{enc} $$ where $Q_{enc}$ is the enclosed charge and $\epsilon_0$ is the permittivity of free space [3](#page=3). ### 2.4 Electric fields using Gauss's Theorem Gauss's theorem provides a simpler method to calculate electric fields for charge distributions with high symmetry (spherical, cylindrical, planar). #### 2.4.1 Field due to an infinite line of charge For an infinite line of charge with linear charge density $\lambda$, a cylindrical Gaussian surface of radius $r$ and length $L$ is used. The electric field is radial and constant in magnitude on the curved surface. $$ E = \frac{2k\lambda}{r} $$ The electric field is directed radially outwards for a positive charge [3](#page=3). #### 2.4.2 Field due to a thin conducting sheet For a thin, infinite conducting sheet with surface charge density $\sigma$, the electric field is uniform and directed perpendicular to the sheet. $$ E = \frac{\sigma}{2\epsilon_0} = \frac{k\sigma}{2} $$ This result holds for regions on both sides of the sheet. For a non-conducting sheet, the field is $E = \sigma/(2\epsilon_0)$ [3](#page=3). #### 2.4.3 Field between two large, oppositely charged sheets When two large, oppositely charged sheets are placed close together, the electric field between them is uniform. If the surface charge densities are $+\sigma$ and $-\sigma$, the field between the plates is: $$ E = \frac{\sigma}{\epsilon_0} = 2k\sigma $$ The field outside the plates is zero due to cancellation. This applies to both non-conducting and conducting sheets [3](#page=3). #### 2.4.4 Field due to a conducting sphere or hollow sphere * **Outside a conducting sphere** of radius $R$ and charge $Q$, at a distance $r > R$: The electric field is the same as that of a point charge $Q$ located at the center. $$ E_{out} = \frac{kQ}{r^2} $$ [3](#page=3). * **On the surface of a conducting sphere** of radius $R$ and charge $Q$: $$ E_{surface} = \frac{kQ}{R^2} $$ [3](#page=3). * **Inside a conducting sphere** ($r < R$): The electric field is zero. $$ E_{in} = 0 $$ [3](#page=3). #### 2.4.5 Field due to concentric conducting spheres For multiple concentric conducting spheres with charges $Q_1, Q_2, \dots$ and radii $R_1, R_2, \dots$: * For $r < R_1$: $E_1 = 0$ (inside the innermost conductor) [4](#page=4). * For $R_1 < r < R_2$: The field depends on the charge enclosed within radius $r$, which is $Q_1$. $E_2 = \frac{k Q_1}{r^2}$ [4](#page=4). * For $R_2 < r < R_3$: The field depends on the total charge enclosed within radius $r$, which is $Q_1 + Q_2$. $E_3 = \frac{k (Q_1 + Q_2)}{r^2}$ [4](#page=4). #### 2.4.6 Field due to a non-conducting sphere of uniform density For a non-conducting sphere of radius $R$ and total charge $Q$ (uniform volume charge density $\rho = Q / (\frac{4}{3}\pi R^3)$): * **Outside the sphere** ($r > R$): The field is the same as that of a point charge $Q$ at the center. $$ E_{out} = \frac{kQ}{r^2} $$ [4](#page=4). * **On the surface of the sphere** ($r = R$): $$ E_{surface} = \frac{kQ}{R^2} $$ [4](#page=4). * **Inside the sphere** ($r < R$): The electric field depends on the charge enclosed within radius $r$. $Q_{enc} = \rho \cdot \frac{4}{3}\pi r^3 = \frac{Q}{\frac{4}{3}\pi R^3} \cdot \frac{4}{3}\pi r^3 = Q \frac{r^3}{R^3}$. $$ E_{in} = \frac{k Q_{enc}}{r^2} = \frac{k (Q \frac{r^3}{R^3})}{r^2} = \frac{kQr}{R^3} $$ [4](#page=4). #### 2.4.7 Field due to a non-conducting sphere with a cavity For a solid non-conducting sphere of radius $R$ and uniform charge density $\rho$, with a spherical cavity of radius $a$, the electric field at any point can be found by superposition. The field is the vector difference between the field of the solid sphere without the cavity and the field of a sphere of density $-\rho$ filling the cavity. * **Field at a point P:** Let the center of the large sphere be O and the center of the cavity be O'. $\vec{E}_{net} = \vec{E}_{sphere\ O} + \vec{E}_{sphere\ O'}$ (where the second sphere has density $-\rho$). If the cavity is centered at O, then $\vec{E}_{net} = \vec{E}_{solid\ sphere}$. If the cavity is offset, say by $\vec{d}$ from the center O, then $\vec{E}_{net}(\vec{r}) = \vec{E}(\vec{r}, \rho, R) + \vec{E}(\vec{r}-\vec{d}, -\rho, a)$. $\vec{E}_{net}(\vec{r}) = \frac{\rho \vec{r}}{3\epsilon_0} - \frac{\rho (\vec{r}-\vec{d})}{3\epsilon_0} = \frac{\rho \vec{d}}{3\epsilon_0} = \frac{\rho \vec{d}}{3\epsilon_0} = \frac{\vec{E}_{solid \ uniform\ field}}{3}$ [4](#page=4). The net field is uniform and directed along the line connecting the centers of the sphere and the cavity. #### 2.4.8 Field due to a non-conducting sphere of varying density If the volume charge density of a non-conducting sphere varies with distance from the center, for example, $\rho(r) = ar^n$: * **Outside the sphere** ($r > R$): First, calculate the total charge $Q = \int_0^R \rho(r) 4\pi r^2 dr$. Then $E_{out} = \frac{kQ}{r^2}$. * **Inside the sphere** ($r < R$): Calculate the enclosed charge $Q_{enc}(r) = \int_0^r \rho(r') 4\pi r'^2 dr'$. Then $E_{in} = \frac{kQ_{enc}(r)}{r^2}$ [4](#page=4). For $\rho = ar^2$, $E_{out} = \frac{Pa R^4}{4r^2}$ and $E_{sur} = \frac{Pa R^2}{4}$ [4](#page=4). ### 2.5 Electrostatic pressure Electrostatic pressure arises from the electric field at the surface of a conductor. The force on a charge element $dq$ on the surface due to the field from all other charges is $dF = dq E_{other}$. At the surface of a conductor, the electric field is $E_{surface} = \sigma/\epsilon_0$. The field due to other charges is $E_{other} = E_{surface}/2 = \sigma/(2\epsilon_0)$. The electrostatic pressure $P_{elec}$ is given by: $$ P_{elec} = \frac{dF}{dA} = \frac{dq E_{other}}{dA} = \frac{\sigma dA \cdot \sigma/(2\epsilon_0)}{dA} = \frac{\sigma^2}{2\epsilon_0} $$ This can also be written in terms of the electric field at the surface: $$ P_{elec} = \frac{1}{2} \epsilon_0 E_{surface}^2 $$ [3](#page=3). ### 2.6 Electrostatic shielding Electrostatic shielding occurs when a conductor encloses a region, preventing external electric fields from penetrating it. For a conductor, any net charge resides on its outer surface. If a conductor with a cavity is present, the electric field inside the cavity is zero, provided there are no charges within the cavity itself. This is because the charges on the conductor redistribute themselves in such a way as to cancel out any external field inside the cavity [4](#page=4). * **For a conductor enclosing a charge $Q_{in}$:** The induced charge on the inner surface is $-Q_{in}$. The net charge on the outer surface is $Q_{outer} = Q_{total} - Q_{in}$ [4](#page=4). * **Electric field inside the cavity:** If there are no charges inside the cavity ($Q_{in}=0$), the field inside the cavity is zero, regardless of any external charges or fields [4](#page=4). * **Uniform electric field inside a conductor:** If a conductor is placed in a uniform external electric field $\vec{E}$, the charges redistribute on the surface such that the net electric field inside the conductor becomes zero. The field lines are distorted around the conductor but do not penetrate it [4](#page=4). --- # Electric dipoles This section introduces electric dipoles, describes how to calculate the electric fields they produce, and explains their behavior in uniform electric fields [2](#page=2). ### 3.1 Definition of an electric dipole An electric dipole consists of two point charges of equal magnitude and opposite sign, separated by a small distance. The dipole moment ($p$) is defined as the product of the magnitude of one of the charges ($q$) and the distance ($2l$) separating them, with its direction pointing from the negative to the positive charge [2](#page=2). $$p = q \times 2l$$ [2](#page=2). ### 3.2 Electric field due to a dipole The electric field produced by an electric dipole can be calculated at different positions relative to the dipole [2](#page=2). #### 3.2.1 Electric field on the axial line On the axial line (along the line joining the two charges), the electric field ($E_{axial}$) is given by [2](#page=2): $$E_{axial} = \frac{2K p}{r^3}$$ [2](#page=2). where $K$ is Coulomb's constant and $r$ is the distance from the center of the dipole to the point on the axial line. For a short dipole, this can also be expressed as $E_{axial} = \frac{2Kp}{r^3}$ or $\frac{2}{2\epsilon_0} \frac{p}{r^3}$ [2](#page=2). #### 3.2.2 Electric field on the equatorial line On the equatorial line (perpendicular to the dipole axis and passing through its center), the electric field ($E_{equatorial}$) is given by [2](#page=2): $$E_{equatorial} = \frac{K p}{r^3}$$ [2](#page=2). Alternatively, $E_{equatorial} = \frac{1}{4\pi\epsilon_0} \frac{p}{r^3}$ [2](#page=2). #### 3.2.3 Electric field at any point At any arbitrary point, the electric field ($E$) due to a dipole can be found using [2](#page=2): $$E = \frac{K p}{r^3} \sqrt{1 + 3\cos^2\theta}$$ [2](#page=2). where $\theta$ is the angle between the dipole moment vector and the line connecting the center of the dipole to the point. The direction of the electric field at any point can be found using $\tan\phi = \frac{1}{2}\tan\theta$ where $\phi$ is the angle the resultant field makes with the dipole axis [2](#page=2). ### 3.3 Dipole placed in a uniform electric field When an electric dipole is placed in a uniform external electric field ($\vec{E}$), it experiences a torque ($\vec{\tau}$) that tends to align it with the field [2](#page=2). #### 3.3.1 Torque on a dipole The torque experienced by a dipole with dipole moment $p$ in a uniform electric field $E$ is given by [2](#page=2): $$\vec{\tau} = \vec{p} \times \vec{E}$$ [2](#page=2). The magnitude of the torque is $\tau = pE\sin\theta$ where $\theta$ is the angle between $\vec{p}$ and $\vec{E}$ [2](#page=2). * **Case 1:** When $\theta = 0^\circ$ (dipole aligned with the field) or $\theta = 180^\circ$ (dipole anti-aligned with the field), the torque is zero ($\tau = 0$) [2](#page=2). * **Case 2:** When $\theta = 90^\circ$, the torque is maximum ($\tau_{max} = pE$) [2](#page=2). #### 3.3.2 Work done to rotate a dipole The work done ($W$) in rotating a dipole from an initial angle $\theta_1$ to a final angle $\theta_2$ in a uniform electric field is [2](#page=2): $$W = pE(\cos\theta_1 - \cos\theta_2)$$ [2](#page=2). This work done is also equal to the change in potential energy ($\Delta U$) of the dipole [2](#page=2). #### 3.3.3 Potential energy of a dipole The absolute potential energy ($U$) of an electric dipole in a uniform electric field is given by [2](#page=2): $$U = -pE\cos\theta$$ [2](#page=2). or $U = -\vec{p} \cdot \vec{E}$ [2](#page=2). * **Case 1:** When $\theta = 0^\circ$, $U = -pE$ which is the condition for stable equilibrium [2](#page=2). * **Case 2:** When $\theta = 180^\circ$, $U = +pE$ which is the condition for unstable equilibrium [2](#page=2). #### 3.3.4 Force on a dipole in a non-uniform electric field A net force acts on a dipole only when it is placed in a non-uniform electric field. The net force ($F_{net}$) is given by [2](#page=2): $$F_{net} = p \frac{dE}{dr}$$ [2](#page=2). where $\frac{dE}{dr}$ represents the gradient of the electric field along the direction of the dipole moment [2](#page=2). --- # Gauss's Theorem and its applications Gauss's Theorem provides a powerful method for calculating electric fields, particularly for symmetrical charge distributions. ### 4.1 Gauss's theorem Gauss's Theorem relates the electric flux through a closed surface to the net electric charge enclosed within that surface. Mathematically, it is expressed as : $$ \oint \vec{E} \cdot d\vec{A} = \frac{q_{enc}}{\epsilon_0} $$ Where: - $ \oint \vec{E} \cdot d\vec{A} $ is the electric flux through the closed surface . - $ \vec{E} $ is the electric field vector . - $ d\vec{A} $ is an infinitesimal area vector on the closed surface, pointing outward . - $ q_{enc} $ is the net electric charge enclosed by the surface . - $ \epsilon_0 $ is the permittivity of free space . This equation states that the total electric flux out of any closed surface is proportional to the total electric charge enclosed within that surface . ### 4.2 Applications of Gauss's theorem Gauss's Theorem is particularly useful for calculating electric fields when the charge distribution possesses a high degree of symmetry (spherical, cylindrical, or planar). This symmetry allows us to choose a Gaussian surface where the electric field is either constant in magnitude and perpendicular to the surface, or parallel to the surface and thus has zero flux through it . #### 4.2.1 Electric field due to a line of charge For an infinitely long line of charge with a uniform linear charge density $ \lambda $, we can use a cylindrical Gaussian surface coaxial with the line of charge. The electric field will be radial and its magnitude will depend only on the distance $ r $ from the line . The electric field magnitude is given by: $$ E = \frac{2k\lambda}{r} $$ Or in terms of $ \epsilon_0 $: $$ E = \frac{\lambda}{2\pi\epsilon_0 r} $$ This shows that the electric field strength decreases inversely with the distance from the line of charge . #### 4.2.2 Electric field due to a thin conducting sheet For a thin, infinite conducting sheet with uniform surface charge density $ \sigma $, the electric field is uniform and directed perpendicular to the sheet. Using a cylindrical Gaussian surface that pierces the sheet, we find the electric field magnitude : $$ E = \frac{\sigma}{2\epsilon_0} $$ > **Tip:** For a non-conducting sheet with uniform charge density, the result is the same . If considering the electric field between two large, oppositely charged conducting sheets with surface charge densities $ +\sigma $ and $ -\sigma $, the electric field in the region between the sheets is: $$ E = \frac{\sigma}{\epsilon_0} $$ And the field outside this region is zero, assuming the sheets are infinitely large . #### 4.2.3 Electric field due to a conducting sphere For a conducting sphere of radius $ R $ with a total charge $ Q $, the charge resides entirely on its surface . - **Outside the sphere ($ r > R $):** The electric field is identical to that of a point charge $ Q $ located at the center of the sphere: $$ E = \frac{kQ}{r^2} $$ - **On the surface of the sphere ($ r = R $):** The electric field is: $$ E_{surface} = \frac{kQ}{R^2} $$ - **Inside the sphere ($ r < R $):** For a conductor in electrostatic equilibrium, the electric field inside is zero: $$ E_{in} = 0 $$ #### 4.2.4 Electric field due to a non-conducting sphere of uniform density For a non-conducting solid sphere of radius $ R $ with a total charge $ Q $ and uniform volume charge density $ \rho $, the charge is distributed throughout its volume . - **Outside the sphere ($ r \ge R $):** Similar to the conducting sphere, the electric field outside is that of a point charge $ Q $ at the center: $$ E_{out} = \frac{kQ}{r^2} $$ - **Inside the sphere ($ r \le R $):** The enclosed charge depends on $ r $. If $ Q $ is the total charge, then $ \rho = \frac{Q}{\frac{4}{3}\pi R^3} $. The enclosed charge is $ q_{enc} = \rho \frac{4}{3}\pi r^3 $. $$ E_{in} = \frac{k(q_{enc})}{r^2} = \frac{k (\rho \frac{4}{3}\pi r^3)}{r^2} = \frac{k \rho \frac{4}{3}\pi r^3}{r^2} $$ Substituting $ k = \frac{1}{4\pi\epsilon_0} $: $$ E_{in} = \frac{\frac{1}{4\pi\epsilon_0} \rho \frac{4}{3}\pi r^3}{r^2} = \frac{\rho r}{3\epsilon_0} $$ This shows that the electric field inside increases linearly with $ r $ . Alternatively, expressing in terms of total charge $ Q $: $$ E_{in} = \frac{kQr}{R^3} $$ #### 4.2.5 Electric field due to concentric conducting spheres For a system of concentric conducting spheres, the charge distribution and electric field can be determined using Gauss's Law and the properties of conductors. The electric field between the shells will depend on the enclosed charge. For example, if there's an inner shell with charge $ Q_1 $ and an outer shell with charge $ Q_2 $ : - Inside the inner shell ($ r < r_1 $): $ E = 0 $ . - Between the inner and outer shells ($ r_1 < r < r_2 $): $ E = \frac{k(Q_1)}{r^2} $ . - Outside the outer shell ($ r > r_2 $): $ E = \frac{k(Q_1 + Q_2)}{r^2} $ . #### 4.2.6 Electric field due to a non-conducting sphere with a cavity Gauss's theorem can also be applied to calculate the electric field in a non-conducting sphere with a cavity. The electric field can be found by considering the superposition of two uniform charge distributions: one for the solid sphere without a cavity, and another for a sphere with a charge density opposite to that of the original sphere, filling the cavity region. The net electric field is the vector sum of the fields due to these two distributions . #### 4.2.7 Electric field due to a non-conducting sphere of varying density If the charge density of a non-conducting sphere varies with the radial distance $ r $, Gauss's Law can still be applied, but the calculation of the enclosed charge $ q_{enc} $ becomes more involved, requiring integration of the variable density over the volume. For a density varying as $ \rho(r) = Cr^n $ : - Outside the sphere ($ r \ge R $): The field is $ E_{out} = \frac{Q_{total}}{4\pi\epsilon_0 r^2} $, where $ Q_{total} $ is the total charge integrated over the sphere's volume . - Inside the sphere ($ r \le R $): $ E_{in} = \frac{1}{4\pi\epsilon_0 r^2} \int_0^r \rho(r') 4\pi r'^2 dr' $ . --- ## Common mistakes to avoid - Review all topics thoroughly before exams - Pay attention to formulas and key definitions - Practice with examples provided in each section - Don't memorize without understanding the underlying concepts

| Term | Definition | |------|------------| | Electrostatics | The branch of physics that studies stationary electric charges and their interactions. | | Electron | A stable subatomic particle with a negative elementary electric charge. | | Coulomb (C) | The SI unit of electric charge, defined as the charge transported by a current of one ampere in one second. | | Electrostatic unit (esu) | An older, cgs unit of electric charge, approximately equal to 3.3356 × 10⁻¹⁰ coulombs. | | Coulomb's Law | A fundamental law in physics that describes the electrostatic interaction between electrically charged particles. The magnitude of the force between two point charges is directly proportional to the product of the magnitudes of charges and inversely proportional to the square of the distance between them. | | Permittivity of free space (${\epsilon}_0$) | A physical constant that is a measure of the capability of the vacuum to permit electric field lines. Its value is approximately $8.852 \times 10^{-12} \text{ F/m}$. | | Dielectric constant (K) | A dimensionless quantity representing the factor by which the electric force between two charges is decreased when a dielectric medium is placed between them, relative to vacuum. It is also known as relative permittivity (${\epsilon}_r$). | | Electric field (E) | A region around a charged particle or object within which a force would be exerted on other charged particles or objects. It is defined as the electric force per unit charge. | | Uniform electric field | An electric field where the electric field strength and direction are the same at all points in the region. | | Work done (W) | The energy transferred when an electric force moves a charge through a distance. It is calculated as the dot product of force and displacement, or as the change in kinetic energy. | | Power (P) | The rate at which work is done or energy is transferred. In electrostatics, it is the product of the instantaneous force and velocity of a charged particle. | | Null point | A point in an electric field where the net electric field strength is zero, typically found between two like charges. | | Electric dipole | A pair of equal and opposite electric charges separated by a small distance. | | Electric dipole moment (p) | A vector quantity representing the strength and orientation of an electric dipole. Its magnitude is the product of the charge magnitude and the distance between the charges, and its direction is from the negative to the positive charge. | | Axial electric field | The electric field at a point lying on the axis of an electric dipole. | | Equatorial electric field | The electric field at a point lying on the perpendicular bisector of an electric dipole. | | Torque ($\tau$) | A twisting force that tends to cause rotation. For a dipole in a uniform electric field, torque tends to align the dipole with the field. | | Potential Energy (U) | The energy stored by an object due to its position relative to some zero point. For a dipole in an electric field, it depends on the orientation of the dipole relative to the field. | | Linear charge density ($\lambda$) | The electric charge per unit length of a conductor. | | Surface charge density ($\sigma$) | The electric charge per unit area of a surface. | | Volume charge density ($\rho$) | The electric charge per unit volume of a region. | | Electric flux ($\Phi$) | A measure of the electric field passing through a given surface. It is calculated as the dot product of the electric field vector and the area vector. | | Gauss's Theorem | A fundamental law in electromagnetism that relates the electric flux through a closed surface to the total electric charge enclosed within that surface. Mathematically, $\oint \vec{E} \cdot d\vec{A} = \frac{Q_{enclosed}}{\epsilon_0}$. | | Conducting sphere | A sphere made of a conductive material, where electric charges can move freely. | | Non-conducting sphere | A sphere made of an insulating material, where electric charges are fixed in position. | | Electrostatic pressure | The pressure exerted by an electric field on a charged surface, which tends to push it outwards. | | Electrostatic shielding | The phenomenon where a conductor shields its interior from external electric fields, resulting in zero electric field inside the conductor. |