Equipotential Surfaces 

1.0What are Equipotential Surfaces?

An equipotential surface is a surface on which the electric potential remains constant at every point.

• If a test charge is moved on an equipotential surface, no work is done because the potential difference between any two points on the surface is zero.
• These surfaces help visualize electric potential in a region, similar to how contour lines represent height on a map.

Equipotential Surface

(1) EPS due to point charge -
EPS of point charge is spherical in shape.

(2) EPS due to line charge -
EPS of line charge is cylindrical in shape.

(3) EPS due to large plane charged sheet -
EPS of large plane sheet is plane in shape.

(4) EPS due to unlike charges -

(5) EPS due to like charges -

Mathematically:
If $(V=\text{constant})$ at all points on a surface, it is an equipotential surface.

2.0Work Done in Equipotential Surface

When a charge $(q)$ moves along an equipotential surface, the electric potential difference between any two points on the surface is zero:

$[\Delta V=0]$

The work done $(W)$ by the electric field is defined as:

$[W=-q,\Delta V]$

Substituting $(\Delta V=0)$:

$[W=-q\cdot 0=0]$

Key Points:

• No work is done when a charge moves along an equipotential surface, regardless of the distance moved.
• This is because the force is always perpendicular to the direction of displacement along the surface.

$[W=q(V_{\text{initial}}-V_{\text{final}})]$

• In contrast, moving a charge between different equipotential surfaces involves work proportional to the potential difference:
  

Illustration:

• Moving a charge along a spherical equipotential surface around a point charge $\to$ work done $=0$.
• Moving the same charge radially outward or inward $\to$ work done $=(q\Delta V\ne 0)$.

This principle is frequently used in JEE problems to quickly determine work done without detailed force integration.

3.0Key Properties of Equipotential Surfaces

1. Work Done is Zero: No work is done in moving a charge from one point to another on an equipotential surface. This is a direct consequence of the definition.
2. Perpendicular to Electric Field Lines: The electric field lines are always perpendicular to the equipotential surfaces at every point. The reason is that if the field had a component parallel to the surface, work would be done in moving a charge along the surface, which contradicts the definition.

3. Cannot Intersect: Two equipotential surfaces can never intersect each other. If they did, the point of intersection would have two different values of electric potential, which is impossible.
4. Closer Spacing Indicates Stronger Field: In regions where the electric field is strong, the equipotential surfaces are closer together. This is because the electric field is related to the potential gradient $(E=-drdV)$. For a given change in potential (dV), a stronger field (E) implies a smaller distance (dr).
5. Spherical Symmetry for a Point Charge: For a single point charge, the equipotential surfaces are concentric spheres centered on the charge.

4.0Equipotential Surfaces due to Different Charge Distributions

Equipotential Surfaces due to a Point Charge

• For a point charge, equipotential surfaces are spherical with the charge at the center.
• All points equidistant from the charge have the same potential.

Equipotential Surfaces due to a Uniform Electric Field

• In a uniform electric field, equipotential surfaces are equally spaced parallel planes perpendicular to the field lines.
• Potential decreases uniformly along the field direction.

Equipotential Surfaces due to a Dipole

• For an electric dipole, equipotential surfaces have a complex pattern due to the interaction of positive and negative charges.
• Surfaces are symmetric about the dipole axis and more closely spaced near the charges.

5.0Relation between Electric Field and Equipotential Surfaces

The electric field vector (E) at any point is always normal to the equipotential surface passing through that point. The magnitude of the electric field is the rate of change of electric potential with distance, known as the potential gradient.

$E=-\frac{dV}{dl}$

The negative sign indicates that the electric field points in the direction of decreasing potential. This relationship is crucial for solving problems where the potential is given as a function of position.

6.0Applications of Equipotential Surfaces

• To determine electric field distribution in complex systems.
• Helps reduce computation of work done in electrostatics.
• Useful in designing capacitors and electrostatic shielding systems.
• Applied in medical equipment such as electrocardiography (ECG) and other electric field mapping technologies.

7.0Solved Examples on Equipotential Surfaces

1. A point charge of $(+2,\mu C)$ is placed at the origin. What is the nature of equipotential surfaces?

Solution:

• Since it’s a point charge, equipotential surfaces are concentric spheres centered at the origin.
• The potential at distance $(r)$ is:

$[V=\frac{1}{4\pi {ϵ}_0}\frac{q}{r}]$

2. What work is done in moving a charge of $(2,C)$ along an equipotential surface where potential is constant?

Solution:

• Work done $(W=q\Delta V)$
• Since $(\Delta V=0)$, $[W=0]$
• No work is done.

3. A charged particle with charge $(q=1.4,\text{mC})$ moves a distance of $(0.4,\text{m})$ along an equipotential surface of $(10,\text{V})$. Calculate the work done by the field during this motion.

Solution:

• Since the particle moves along an equipotential surface, the potential difference $(\Delta V=0)$.
• The work done $(W)$ is given by:
  $[W=-q\Delta V]$
• Substituting $(\Delta V=0)$: $[W=-q\times 0=0]$
  
• Answer: The work done is zero.

4. Calculate the distance between two equipotential surfaces differing by $(10,\text{V})$ in a uniform electric field of $(200,\text{V/m})$.

Solution:

• The potential difference $(\Delta V=10,\text{V})$ and the electric field $(E=200,\text{V/m})$.
• The relationship between potential difference and electric field is:
  $[\Delta V=E\times d]$
• Solving for $(d)$: $[d=\frac{\Delta V}{E}=\frac{10}{200}=0.05,\text{m}]$
  
• Answer: The distance between the equipotential surfaces is 0.05 meters.

5. A charge $(q=2,\mu \text{C})$ is moved from a point at potential $(V_1=5,\text{V})$ to another point at potential $(V_2=15,\text{V})$. Calculate the work done by the electric field.

Solution:

• The work done $(W)$ is given by:

$[W=-q\Delta V=-q(V_2-V_1)]$

• Substituting the values:

$[W=-2\times 10^{-6}\times (15-5)=-2\times 10^{-6}\times 10=-20\times 10^{-6},\text{J}]$

$[W=-20,\mu \text{J}]$

• Answer: The work done is $-20\mu \mathrm{J}$.

6. Given two equipotential surfaces at potentials $(V_1=20,\text{V})$ and $(V_2=10,\text{V})$, separated by a distance of $(0.2,\text{m})$, calculate the electric field between them.

Solution:

• The potential difference $(\Delta V=V_1-V_2=20-10=10,\text{V})$.
• The distance between the surfaces $(d=0.2,\text{m})$.
• The electric field $(E)$ is given by: $[E=\frac{\Delta V}{d}=\frac{10}{0.2}=50,\text{V/m}]$
• Answer: The electric field between the surfaces is 50 V/m.

7. A charge in moving via three paths, 1, 2, and 3. as shown in figure. Find work done in each case.

Solution:
Potential at points A, B, C and D is

$V=\frac{kq}{R}$

All the points are equipotential points, so work done in each case is zero.

8. Find relation between $E_A\text{ and }{E}_B$.

Solution:

$|E|=\frac{\Delta V}{d}$and here $\Delta \mathrm{V}\Delta \mathrm{V}$ is same, but $dd$ is less near point A so $E_A$​ will be greater than $E_B$.

9. Concentric spherical equipotential surfaces due to a point charge are shown in the diagram, if: $V_1-V_2=V_2-V_3$ then –

Solution:
If electric field between $V_1$ and $V_2$ be the $E_1$​ and between $V_2$​ and $V_3$​ be the $E_2$​ then we can write (using approximation)

$V_1-V_2=E_1(x)$

$V_2-V_3=E_2(y)$

We can see here $E_1>E_2$ so according to given condition that $V_1-V_2=V_2-V_3$ we can conclude that $x<y$.

10. Find which has more work done in moving a charge from A to B?

Solution:

Use $\mathrm{W}=\mathrm{q}(\Delta \mathrm{V})$

$W_1>W_2=W_3$

8.0Frequently Asked Questions (FAQs)

Q1. Why is no work done on an equipotential surface?
Because the potential difference between any two points is zero, hence no work is required to move a charge.

Q2. What is the shape of equipotential surfaces for a point charge?
They are spherical, centered at the charge.

Q3. How are equipotential surfaces related to electric field lines?
They are always perpendicular to electric field lines.

Q4. What is the significance of closely spaced equipotential surfaces?
It indicates a region of strong electric field.

Q5. Can two equipotential surfaces intersect?
No, because it would mean a single point has two different potentials, which is impossible.