Electrostatic Potential and Capacitance: Comprehensive NEET Physics Notes

1. Introduction to Electrostatic Potential

1.1 Electrostatic Potential Energy

In previous classes, potential energy was introduced as the energy stored when an external force does work against a force, such as gravitational or spring force. The Coulomb force between two stationary charges is also a conservative force. Like gravitational potential energy, we can define the electrostatic potential energy of a charge in an electrostatic field.

Imagine bringing a test charge $q$ from point $R$ to point $p$ against the repulsive force due to a charge $Q$ at the origin. If both $Q$ and $q$ are positive, an external force equal to the negative of the electric force is applied, ensuring no net force or acceleration on $q$. The work done by the external force is stored as potential energy. Thus, the work done in moving a charge $q$ from $R$ to $p$ is given by:

$W_{RP}={\int }_R^P{\mathbf{F}}_{ext}\cdot d\mathbf{r}=-{\int }_R^P{\mathbf{F}}_E\cdot d\mathbf{r}$

The potential energy difference between points $p$ and $R$ is:

$\Delta U=U_P-U_R=W_{RP}$​

Did You Know?

The concept of potential energy is only meaningful for conservative forces, where the work done is path-independent.

Mnemonic:

"CAP" - Conservative forces, like Coulomb and gravitational forces, have Associated Potential energy.

1.2 Electrostatic Potential

Electrostatic potential $V$at a point is defined as the work done per unit charge in bringing a positive test charge from infinity to that point. For a charge configuration, the potential difference between points $P$ and $R$ is:

$V_P-V_R=-{\int }_R^P\mathbf{E}\cdot d\mathbf{r}$

If we choose the potential to be zero at infinity, the potential $V$ at a point due to a charge $Q$ at the origin is:

$V(r)=\frac{Q}{4\pi {ϵ}_{0}r}$

Real-life Application:

Electrostatic potential is crucial in designing electronic components like capacitors and transistors, which are the building blocks of modern electronic devices.

NEET Tip:

Always remember that the potential at infinity is taken as zero unless otherwise specified.

Quick Recap:

• Electrostatic potential energy is the energy stored due to the position of charges in an electrostatic field.
• Electrostatic potential is the work done per unit charge in bringing a test charge from infinity to a point in space.
• The potential due to a point charge $Q$ at a distance $r$ is given by $V=\frac{Q}{4\pi {ϵ}_{0}r}$.

2. Potential Due to Different Charge Distributions

2.1 Potential Due to a Point Charge

For a point charge $Q$ at the origin, the potential $V$ at a distance $r$ is:

$V(r)=\frac{Q}{4\pi {ϵ}_{0}r}$

This expression holds for both positive and negative charges, with the potential being positive for positive charges and negative for negative charges.

NEET Problem-Solving Strategy:

For calculating potential due to multiple point charges, use the principle of superposition, adding the potentials due to individual charges algebraically.

2.2 Potential Due to an Electric Dipole

An electric dipole consists of charges $+q$ and $-q$separated by a distance $2a$. The potential at a point $P$ with position vector $r$ due to a dipole is:

$V(\mathbf{r})=\frac{1}{4\pi {ϵ}_0}\frac{\mathbf{p}\cdot \mathbf{r}}{r^3}$

where $\mathbf{p}=q\cdot 2a$ is the dipole moment vector.

Common Misconception:

The potential due to a dipole decreases as $1/r^2$ at large distances, not as$1/r$like a single charge.

Quick Recap:

• The potential due to a point charge $Q$ is ​$V=\frac{Q}{4\pi {ϵ}_{0}r}$
• The potential due to a dipole at a point is $V=\frac{1}{4\pi {ϵ}_0}\frac{\mathbf{p}\cdot \mathbf{r}}{r^3}$​.

3. Equipotential Surfaces

3.1 Concept and Properties

Equipotential surfaces are surfaces where the potential is constant. For a point charge, these surfaces are concentric spheres centered on the charge. For a uniform electric field, they are planes perpendicular to the field lines.

3.2 Relation Between Field and Potential

The electric field is always perpendicular to equipotential surfaces and points in the direction of decreasing potential. The magnitude of the electric field $E$ is related to the potential difference ΔV and the distance$\Delta l$ by:

$E=-\frac{\Delta V}{\Delta l}$

Did You Know?

No work is done in moving a charge along an equipotential surface, as the potential difference is zero.

Concept Connection:

In chemistry, the concept of equipotential surfaces is similar to the idea of contour lines on a map, where each line represents a constant potential energy.

Quick Recap:

• Equipotential surfaces are perpendicular to electric field lines.
• The electric field strength is related to the rate of change of potential with distance.

4. Capacitance

4.1 Capacitors and Capacitance

A capacitor is a device that stores electric charge and energy. The capacitance $C$ is defined as the ratio of the charge $Q$ on one plate to the potential difference $V$ between the plates:

$C=\frac{Q}{V}$

The unit of capacitance is the farad (F).

4.2 Parallel Plate Capacitor

For a parallel plate capacitor with plate area$A$ and separation $d$, the capacitance is:

$C=\frac{{ϵ}_{0}A}{d}$

where ${ϵ}_0$​ is the permittivity of free space.

Real-life Application:

Capacitors are used in various electronic circuits, including filters, timers, and memory storage devices.

NEET Exam Strategy:

In problems involving capacitors, ensure to account for the dielectric constant if a dielectric material is present between the plates.

Quick Recap:

• Capacitance is the ability of a system to store charge per unit potential difference.
• The capacitance of a parallel plate capacitor is directly proportional to the plate area and inversely proportional to the separation between the plates.

Practice Questions

1. Calculate the potential at a point 5 cm away from a charge of $2\times 10^{-7}\text{C}$.
2. Two charges, $3\times 10^{<}$