Equipotential Surface

The term equipotential comes from combining two words, ‘equi’, which means ‘equal’, and ‘potential’, which makes it ‘equal potential’ or ‘equipotential’. This term arrives from its usage in vector calculus and topological science, where the word ‘equipotential’ refers to a surface with a constant scalar value. This equipotential of a scalar potential of 3 dimensions will be in a 2-dimensional space. The equipotential region of 3-dimensional scalar potential is called an equipotential surface.

What is an equipotential surface?

In mathematics, an equipotential surface can be defined as the locus of all points in a region with a constant scalar potential. This concept of a surface that arises from a 3-dimensional scalar function is also referred to as a level surface in 2 dimensions. This can be mathematically expressed by using the following method:

Let Φ ( x, y, z ) be a scalar function in the 3-dimensional space,

Let the value of Φ ( x, y, z ) for a certain point in the domain of the function be given as

Φ ( x1, y1, z1 ) = a

Where a is a constant value.

Let us consider another point in the domain of Φ with the same value for Φ let this point be: 

Φ ( x2, y2, z2 ) = a

Similarly, we can find n such points in space in which Φ ( x, y, z ) has the value a.

Let this point be represented by Φ ( xn, yn, zn ) = a

Now we can see that we have a collection of n points in space in which Φ ( x, y, z ) has the same value i.e. a. Therefore, if we now plot all these points in the 3-dimensional space, we get a level surface known as an equipotential surface. 

This concept of an equipotential surface is widely used in many fields of physics.

Properties of an equipotential surface:

• All points on an equipotential surface of a function will have the same value.

• If the function is in the space of ‘n’ dimensions, then the equipotential surface will be of ‘n – 1’ dimensions.

• The direction of change of the equipotential surface of a function Φ ( x, y, z ) will be given by taking the gradient of the function and dividing it by the magnitude of the gradient of the function.

( ▽ Φ ( x, y, z ) ) / | ▽ Φ ( x, y, z ) | = n  

Here n is the unit vector in the direction of change in the equipotential surface. 

Applications of equipotential surface:

• In electrostatics:

In electrostatics, the concept of equipotential surface is used for an electric field due to a static charge.

The scalar potential of the electric field is given by the following equation:

• E = -▽V  

Here V is the electric potential, and E is the electric field. So a locus of points in space with the same potential, V, is called an equipotential surface in electrostatics.

This surface can be of any shape as long as the resultant electric field is perpendicular to the equipotential surface and the electric potential value is the same throughout the surface. 

In the case of an electrical conductor, the conductor’s surface will be an equipotential surface such that any two points on the surface will have the same electrical potential, and if we connect any two points on the surface, no charge will flow due to a lack of electrical potential difference.

For example, for a  point charge, the equipotential surface is usually a sphere with a charge at its centre.

• In mechanics:

In mechanics, an equipotential surface can be used to study gravity. The geopotential equipotential surface can be seen as the sea level for the gravity on earth. The fact that there exists a sea level on earth is a direct consequence of the equipotential surface that arises due to the gravity on earth.   

An object that is stationary on earth will not move left or right on the earth’s surface because of the implication of the equipotential surface on earth. This is because there is no change in gravitational force on either side. But once you lift the object from the surface perpendicularly, the object moves downward because we have moved the object from one equipotential surface to another. This change in equipotential surface results in the accumulation of gravitational potential energy in the object.

• In magnetostatics:

In magnetostatics, the equipotential surface can be governed by the magnetic potential, which is given by the equation:

B = ▽ x A 

Where B is the magnetic field, and A is the magnetic vector potential.

The equipotential surface of a magnetic field in straight is usually a closed-loop perpendicular to the magnetic field.

Conclusion

An equipotential surface is a level surface where all the points in the surface will be of equal potential. An equipotential surface can have any shape. The concept of equipotential is used in many cases like gravitation, electrostatics, magnetostatic, etc. The work done against potential in its equipotential surface will always be zero.