Equipotential Surface: Work Done, Properties & Magnitude

Equipotential Surface is the surface that has a constant value of the electrical potential at all the points on that surface. The formation of an equipotential surface depends on the type of medium i.e. isotropic, non-isotropic, and the amount of charge distribution.

• A single-point charge of the equipotential surface is a concentric spherical surface centered at the charge. 
• An equipotential surface is thus a surface where the potential is the same at every point on the surface.

Read More: Electrostatic Potential and Capacitance

Key Terms: Equipotential, Equipotential Surfaces, Work, Electric Field, Electric Charge, Electric Potential, Work, Electrical potential, Charge

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Equipotential Surfaces

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The surface that forms the locus of all points that are at the same potential forms the equipotential surface. To move a charge from one point to another on the equipotential surface, work is not required. For a single charge q, the potential can be calculated using the following formula. Here, V is constant if r is constant. 

Therefore, equipotential surfaces of a single-point charge are concentric spherically centered at the potential charge.

Equipotential Surface Schematic

Equipotential Points

Equipotential points are those points in an electric field that are at the same electric potential.

Equipotential Line

When equipotential points are connected by a line or curve, it is called an equipotential line.

Equipotential Surface

When equipotential points lie on a surface, it is called an equipotential surface.

Equipotential Volume

If equipotential points are distributed throughout a space or volume, it is called an equipotential volume.

Equipotential Surfaces for a Uniform Electric Field

The below figure shows equipotential surfaces for a uniform electric field

Equipotential Surfaces for an Isolated Point Charge

The figure below shows Equipotential surfaces for an isolated point charge

Equipotential Surfaces for a Pair of Similar Charge

The figure below shows the equipotential surfaces for a pair of similar charge

Equipotential Surfaces for a Pair of Opposite Charge

The equipotential surfaces for two charges of opposite sign are shown below

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Work Done in Equipotential Surface

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The work done in moving a point charge from one point to another in an equipotential surface is zero. If a point charge is moved from point V_Y to V_Z, in an equipotential surface then the work done in the moving point charge can be calculated using the following equation:

W = q₀ (V_Y - V_z)

As the value of V_Y - V_z is zero, the total work done W = 0.

Work Done in Equipotential Surface

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Properties of Equipotential Surface

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The properties possessed by equipotential surfaces are mentioned below:

1. The electric field is always perpendicular to an equipotential surface.
2. Two equipotential surfaces never intersect each other.
3. For a point charge, equipotential surfaces are concentric spherical shells.
4. In a uniform electric field, any plane normal to the field direction is an equipotential surface.
5. For a uniform electric field, equipotential surfaces are planes normal to the x-axis.
6. Inside a hollow-charged spherical conductor, the potential remains constant. It can be considered as an equipotential volume. No work is needed to move a point charge from the center to the surface.
7. The direction of the equipotential surface goes from high potential to low potential.
8. For an isolated charge, the equipotential surface is a sphere. This means concentric spheres around the charge are different equipotential surfaces.
9. The space between equipotential surfaces enables us to find regions having a strong and weak field.

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Electric Equipotential Surface

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If electric field lines are present in an n-dimensional space, then the equipotential surface is perpendicular to this plane. This imaginary surface is along the z-axis if the field is set in an X-Y plane.

• The equipotential surface always remains perpendicular to the electric field lines.
• It stays at an equal potential as there is no change expected on this surface.

Along with the equipotential surface, it is necessary to consider the work done when we move the charge along the surface. In electrostatics, the work done is calculated by:

W = - q × ΔV= - ΔU

Where,

q is the charge

ΔV is the charge in potential

ΔU is the electric potential energy gained by the charge when it is forced to move in external electric potential.

The work done here is at the expense of electric potential. So W = - ΔU

The charge doesn’t gain any energy, as there is no change in electric potential because the surfaces are equipotential. And as there is no change in energy, no work is done.

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Magnitude of Electric Field on Equipotential Surface

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The value of the electric field in the Equipotential surface direction is zero, this is because the integral line of the electrical field is potential. Therefore, for the potential to remain the same, the electrical field must be zero.

Electrical Field on Equipotential Surface

Read More: Electric Field and Charge Important Questions

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Things to Remember

• Surface with constant electrostatic potential values is termed as an equipotential surface.
• Work done in an equipotential field is given by: W = q₀ (V_Y - V_z)
• The electric field is perpendicular to the surface and the surfaces never intersect one another.
• Equipotential surfaces are planes normal to the x-axis.
• In a hollow spherical conductor the potential is zero.
• The electric field value in an equipotential surface is zero. 

Read More: NCERT Solutions for Class 12 Physics Chapter 2 

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Previous Year Questions

1. A conducting sphere of radius R=20cm is given a charge Q….[JIPMER 2015]
2. A metallic sphere is placed in a uniform electric field. The line of force follow the path (s) shown in  ….[VITEEE 2018]
3. On moving a charges , the potential difference between the points is… [VITEEE 2011]
4. The effective capacitance between two points is….
5. Voltage rating of a parallel plate capacitor is….[JEE Main 2018]
6. A bar magnet is10 cm long is kept with its north….[BITSAT 2019]
7. In the circuit shown, find C if the effective capacitance of the whole circuit is….[JEE Main 2019]
8. In the figure shown below, the charge on the left plate of the 10μF capacitor is −30μC….[JEE Main 2019]
9. In The Figure Shown After The Switch S Is Turned from postion a to b….[JEE Main 2019]
10. The effective capacitance of the whole circuit is to…...[JEE Main 2019]
11. The dielectric constant of a material which when fully inserted in above capacitor, gives same capacitance….[JEE Main 2019]
12. A solid conducting sphere, having a charge Q, is surrounded by an uncharged conducting hollow  ….​.[JEE Main 2019]
13. A Plane Electromagnetic Wave Of Frequency 50 MHz travels in….[JEE Main 2019]
14. While a capacitor remains connected to a battery, a dielectric slab is slipped between the plates…..[KCET 2001]
15. The electron is accelerated through a potential difference of 10 V. The additional energy acquired by the electron is…
16. Watt per ampere is unit of ?
17. A Parallel Plate Capacitor With Square Plates Is F….[JEE Main 2019]
18. The energy stored in a capacitor of capacity C and potential V is given by...[NEET 1996]
19. What is the final potential difference across each capacitor? [KEAM]
20. As shown in the figure, charges are placed at the vertices…. [VITEEE 2011]