"Electric Potential and Capacitors"

About

While electric field describes the force aspect of electricity, electric potential describes the energy aspect — it tells us how much work is needed to move a charge. This chapter connects field and potential, introduces the capacitor as a device for storing charge and energy, and explores how dielectrics enhance capacitance.

Key Concepts

16.1 Electric Potential

Electric potential at a point is the work done per unit positive charge to bring it from infinity to that point:

$V=\frac{W}{q_0}$

SI unit: volt (V) = J/C. Potential is a scalar quantity.

Potential due to a point charge: $V=\frac{1}{4\pi {\epsilon }_0}\frac{q}{r}=k\frac{q}{r}$

Potential due to a charged sphere (outside, $r\ge R$): $V=k\frac{q}{r}$

(Behaves as if all charge is concentrated at the centre.)

16.2 Potential Difference

$V_{AB}=V_B-V_A=\frac{W_{A\to B}}{q_0}$

In a uniform electric field: $V=Ed$

16.3 Relation Between E and V

$\vec{E}=-\frac{dV}{dr}$

• Electric field points in the direction of decreasing potential
• In a region of constant potential, $E=0$
• If $E=0$ at a point, $V$ is NOT necessarily zero (e.g., between two equal positive charges, $E=0$ but $V\ne 0$)

16.4 Equipotential Surfaces

Surfaces where potential is the same at every point.

Properties:

• No work is done in moving a charge along an equipotential surface
• Electric field is always perpendicular to equipotential surfaces
• Two equipotential surfaces cannot intersect (potential at a point must be unique)
• For a point charge, equipotential surfaces are concentric spheres

16.5 Capacitance

Capacitance is the ability of a conductor to store charge:

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

SI unit: farad (F) = C/V. Practical units: μF ($10^{-6}$ F), pF ($10^{-12}$ F).

Dimensional formula: $[M^{-1}{L}^{-2}{T}^4{A}^2]$

16.6 Parallel Plate Capacitor

$C=\frac{{\epsilon }_{0}A}{d}$

Where $A$ = plate area, $d$ = separation.

With dielectric (of dielectric constant $K$): $C=K{C}_0=\frac{K{\epsilon }_{0}A}{d}$

Effect of inserting a dielectric:

16.7 Combinations of Capacitors

Series: $\frac{1}{C_{\text{eq}}}=\frac{1}{C_1}+\frac{1}{C_2}+\dots$

• Same charge on each capacitor
• Voltage divides

Parallel: $C_{\text{eq}}=C_1+C_2+\dots$

• Same voltage across each capacitor
• Charge divides

16.8 Energy Stored in a Capacitor

$U=\frac{1}{2}C{V}^2=\frac{1}{2}QV=\frac{Q^2}{2C}$

Energy density (energy per unit volume in electric field): $u=\frac{1}{2}{\epsilon }_0{E}^2$

INTEXT QUESTIONS 16.1

Q1. A metallic sphere of radius R has a charge +q uniformly distributed on its surface. What is the potential at a point r (> R) from the centre of the sphere?

Ans: For points outside a charged sphere, it behaves as if all charge is concentrated at the centre: $V=\frac{1}{4\pi {\epsilon }_0}\frac{q}{r}=\frac{q}{4\pi {\epsilon }_{0}r}$

Q2. Calculate the work done when a point charge is moved in a circle of radius r around a point charge q.

Ans: All points on the circle are equidistant from the central charge → same potential. Work done = $q_0(V_{\text{final}}-V_{\text{initial}})=q_0(V-V)=0$.

Work done = 0.

Q3. The electric potential V is constant in a region. What can you say about the electric field E in this region?

Ans: Since $E=-dV/dr$, if $V$ is constant, $dV/dr=0$, so $E=0$. In a region of constant potential, the electric field is zero.

Q4. If electric field is zero at a point, will the electric potential be necessarily zero at that point?

Ans: No. $E=-dV/dr=0$ only means potential is not changing — it could be at a maximum, minimum, or constant. Example: At the centre between two equal positive charges, $E=0$ but $V\ne 0$.

Q5. Can two equipotential surfaces intersect?

Ans: No. If they intersected, there would be two different potential values at the same point — impossible since potential is unique at every point. Also, $E$ would have two perpendicular directions at the intersection — also impossible.

INTEXT QUESTIONS 16.2

Q1. Write the dimensions of capacitance.

Ans: From $C=Q/V$:

• $[Q]=[AT]$
• $[V]=[M{L}^2{T}^{-3}{A}^{-1}]$

$[C]=\frac{[AT]}{[M{L}^2{T}^{-3}{A}^{-1}]}=[M^{-1}{L}^{-2}{T}^4{A}^2]$

Q2. What is the potential difference between two points separated by a distance d in a uniform electric field E?

Ans: $V=Ed$

Terminal Exercise

1. Define electric potential and potential difference. Derive the expression for potential due to a point charge.

2. Derive the relation between electric field and potential: $\vec{E}=-dV/dr$.

3. What are equipotential surfaces? Draw equipotential surfaces for: (a) a point charge, (b) a uniform electric field, (c) an electric dipole.

4. Define capacitance. Derive the expression for the capacitance of a parallel plate capacitor: $C={\epsilon }_{0}A/d$.

5. Derive expressions for the equivalent capacitance when capacitors are connected in: (a) series, (b) parallel.

6. Show that the energy stored in a capacitor is $U=\frac{1}{2}C{V}^2$. Also derive the expression for energy density.

7. Explain the effect of inserting a dielectric slab between the plates of a capacitor on: (a) capacitance, (b) electric field, (c) potential difference.

8. A parallel plate capacitor has plates of area 0.1 m² separated by 1 mm. Find its capacitance. If a dielectric of $K=5$ is inserted, what is the new capacitance? (${\epsilon }_0=8.85\times 10^{-12}$)

9. Two capacitors of 3 μF and 6 μF are connected in series across a 100 V battery. Find: (a) equivalent capacitance, (b) charge on each, (c) voltage across each.

10. A 10 μF capacitor is charged to 200 V. Find the energy stored. The capacitor is then disconnected and connected across an uncharged 30 μF capacitor. Find the common potential and the energy lost.

11. What is the work done in moving a charge of 5 μC between two points having a potential difference of 100 V?

12. The potential at a point 0.1 m from a point charge is 900 V. Find the magnitude of the charge.

Worked Examples

Example 1: Potential

Problem: Find the potential 2 m from a point charge of 4 μC.

Solution: $V=k\frac{q}{r}=9\times 10^9\times \frac{4\times 10^{-6}}{2}=1.8\times 10^4\text{ V}$

Example 2: Capacitance

Problem: A parallel plate capacitor has plates of area 0.5 m² separated by 2 mm. Find its capacitance.

Solution: $C=\frac{{\epsilon }_{0}A}{d}=\frac{8.85\times 10^{-12}\times 0.5}{2\times 10^{-3}}=2.21\times 10^{-9}\text{ F}=2.21\text{ nF}$

Example 3: Energy

Problem: A 5 μF capacitor is charged to 100 V. Find the stored energy.

Solution: $U=\frac{1}{2}C{V}^2=\frac{1}{2}\times 5\times 10^{-6}\times (100{)}^2=0.025\text{ J}$

Common Mistakes

1. Confusing potential (scalar) with potential energy: $V=U/q$ — potential is per unit charge.
2. Thinking $V$ must be zero where $E=0$: Not true — $E=0$ only means $dV/dr=0$.
3. Using wrong formula for series vs parallel capacitors: Series → reciprocal sum (like resistors in parallel). Parallel → direct sum.
4. Forgetting that charge is same in series capacitors: Only voltage divides.
5. Applying capacitor formulas when dielectric partially fills the gap: Requires careful treatment.

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