Physics: Class 12 (Book 1) — Chapter 2: Electrostatic Potential and Capacitance

The work done $W_{RP}$ by an external force $\mathbf{F}_{ext}$ in bringing a charge $q$ from point $R$ to point $P$ against the repulsive electric force $\mathbf{F}_E$ (where $\mathbf{F}_{ext} = -\mathbf{F}_E$) is given by:

This work done increases the potential energy of the charge by:

• The work done by an electrostatic field depends only on the initial and final positions of the charge, and is completely path-independent .
• The actual value of potential energy is not physically significant; only the difference $\Delta U$ is physically meaningful.
• By convention, the potential energy at infinity is chosen to be zero ($U_{\infty} = 0$). Thus, the potential energy of a charge $q$ at any point $P$ is the work done by an external force in bringing it from infinity to that point: $W_{\infty P} = U_P$.

The potential difference between two points $P$ and $R$ is the work done per unit positive charge in bringing it from $R$ to $P$:

Taking the potential at infinity to be zero ($V_{\infty} = 0$), the electrostatic potential at any point $P$ is defined as:

The work done by an external force in bringing a unit positive charge from infinity to that point without acceleration.

1. The electrostatic force on a unit positive charge at an intermediate point $P'$ at distance $r'$ is: $$F = \frac{Q \times 1}{4\pi\epsilon_0 r'^2} \hat{\mathbf{r}}'$$
2. The work done against this force for a small displacement $dr'$ is: $$dW = -\frac{Q}{4\pi\epsilon_0 r'^2} dr'$$
3. Integrating from infinity ($r' = \infty$) to the point $P$ ($r' = r$): $$W = -\int_{\infty}^{r} \frac{Q}{4\pi\epsilon_0 r'^2} dr' = \frac{Q}{4\pi\epsilon_0 r}$$
4. By definition, this work done is the potential $V(r)$ at distance $r$: $$V(r) = \frac{Q}{4\pi\epsilon_0 r}$$

The potential $V$ at a point $P$ at distance $r$ from the center of the dipole is the sum of potentials due to $+q$ and $-q$:

Using geometry and the approximation $r \gg a$:

• On the dipole axis ($\theta = 0$ or $\theta = \pi$): $$V = \pm \frac{p}{4\pi\epsilon_0 r^2}$$
• On the equatorial plane ($\theta = \pi/2$): $$V = 0$$

• Outside the shell ($r \ge R$): The potential is as if the entire charge $q$ is concentrated at the center: $$V = \frac{1}{4\pi\epsilon_0} \frac{q}{r}$$
• Inside the shell ($r < R$): The electric field inside is zero, meaning the potential is constant and equal to its value at the surface: $$V = \frac{1}{4\pi\epsilon_0} \frac{q}{R}$$

• No work is done in moving a test charge between any two points on an equipotential surface.
• The electric field $\mathbf{E}$ at any point is always normal to the equipotential surface passing through that point.
• For a uniform electric field along the $x$-axis, the equipotential surfaces are planes parallel to the $y$-$z$ plane.

The relation between the electric field $\mathbf{E}$ and potential gradient is given by:

This \implies:

1. The electric field is in the direction in which the potential decreases steepest.
2. Its magnitude is given by the change in potential per unit displacement normal to the equipotential surface.

The potential energy $U$ of a system of two charges $q_1$ and $q_2$ separated by distance $r_{12}$ is the work done in assembling them from infinity:

For three charges $q_1, q_2, q_3$ located at $\mathbf{r}_1, \mathbf{r}_2, \mathbf{r}_3$:

• Single charge $q$ in external potential $V(\mathbf{r})$: $$U = qV(\mathbf{r})$$
• Two charges $q_1, q_2$ in external potential $V(\mathbf{r})$: $$U = q_1 V(\mathbf{r}_1) + q_2 V(\mathbf{r}_2) + \frac{1}{4\pi\epsilon_0} \frac{q_1 q_2}{r_{12}}$$
• Dipole in a uniform external field $\mathbf{E}$: The potential energy $U(\theta)$ of a dipole with moment $\mathbf{p}$ at angle $\theta$ is: $$U(\theta) = -\mathbf{p} \cdot \mathbf{E} = -pE \cos\theta$$

1. Inside a conductor, the electrostatic field is zero.
2. At the surface of a charged conductor, the electrostatic field must be normal to the surface at every point.
3. The interior of a conductor can have no excess charge in the static situation. (All excess charge resides on the outer surface).
4. Electrostatic potential is constant throughout the volume of the conductor and has the same value on its surface.
5. Electric field at the surface of a charged conductor is: $$\mathbf{E} = \frac{\sigma}{\epsilon_0} \hat{\mathbf{n}}$$ where $\sigma$ is the local surface charge density.
6. Electrostatic shielding: The electric field inside a cavity of any conductor is always zero, shielding the interior from external electrical influences.

• Non-polar molecules: The centers of positive and negative charges coincide, resulting in zero intrinsic dipole moment (e.g., $O_2, H_2$).
• Polar molecules: The centers of positive and negative charges are separated, giving them a permanent dipole moment (e.g., $HCl, H_2O$).

The dipole moment per unit volume is called polarisation ($\mathbf{P}$). For linear isotropic dielectrics:

where $\chi_e$ is the electric susceptibility of the dielectric medium.

The charge $Q$ on a capacitor is directly proportional to the potential difference $V$ between the conductors:

The constant $C$ is the capacitance . Its SI unit is the Farad (F) ($1\text{ F} = 1\text{ C V}^{-1}$).

For two parallel plates of area $A$ separated by a small distance $d$ in vacuum:

When the space between the plates is completely filled with a dielectric of dielectric constant $K$:

where $\epsilon = K \epsilon_0$ is the permittivity of the medium.

For $n$ capacitors connected in series, the charge $Q$ is the same on each, and the total potential drop is the sum of individual drops:

For $n$ capacitors connected in parallel, the potential difference $V$ is the same across each, and the total charge is the sum of individual charges:

The work done in charging a capacitor is stored as electrostatic potential energy $U$:

The energy stored per unit volume in the electric field between the plates is the energy density ($u$):