Potential and Potential Difference

Potential and Potential Difference :

The force acting on a charge at infinity is zero, hence ‘infinity’ is chosen as the theoretical place of zero electric potential. Therefore, potential at any point in an electric field may be defined as numerically equal to the work done in bringing a positive charge of one coulomb from infinity to that point against the electric field. The unit of this potential will depend on the unit of charge taken and the work done. If, in shifting one coulomb from infinity to a certain point in the electric field, the work done is one joule, then potential of that ponit is one volt.

Obviously, potential is work per unit charge,

Similarly, potential difference (p.d.) of one volt exists between two points if one joule of work is done in shifting a charge of one coulomb from one point to the other.

What is potential difference ?

Potential difference is the difference in the amount of energy that charge carriers have between two points in a circuit.

Measured in Volts: Potential difference (p.d.) is measured in volts (V) and is also called voltage. The energy is transferred to the electrical components in a circuit when the charge carriers pass through them. We use a voltmeter to measure potential difference (or voltage).

Potential Difference formula: V = I x R

The potential difference (which is the same as voltage) is equal to the amount of current multiplied by the resistance. A potential difference of one Volt is equal to one Joule of energy being used by one Coulomb of charge when it flows between two points in a circuit.

Potential at a Point :

Consider a positive point charge of Q coulombs placed in air. At a point x metres from it, the force on one coulomb positive charge is Q/4 πε₀ x² (Figure given below). Suppose, this one coulomb charge is moved towards Q through a small distance dx. Then, work done is

The negative sign is taken because dx is considered along the negative direction of x.

The total work done in bringing this coulomb of positive charge from infinity to any point D which is d metres from Q is given by

By definition, this work in joules in numerically equal to the potential of that point in volts.

We find that as d increases, V decreases till it becomes zero at infinity.

Potential of a Charged Conducting Sphere :

The above formula V = Q/4πε₀ ε_r d applies only to a charge concentrated at a point. The problem of finding potential at a point outside a charged sphere sounds difficult, because the charge on the sphere is distributed over its entire surface and so, is not concentrated at a point. But the problem is easily solved by nothing that the lines of force of a charged sphere, like A in by noting that the lines of force of a charged sphere, like A in Figure (given below) spread out normally from its surface. If produced backwards, they meet at the centre of A.

Hence for finding the potentials at points outside the sphere, we can imagine the charge on the sphere as concentrated at its centre O. If r is the radius of sphere in metres and Q its charge in coulomb then, potential of its surface is Q/4π ε₀ r volt and electric intensity is Q/4πε₀ r². At any other point ‘d’ metres from the centre of the sphere, the corresponding values are Q/4π ε₀ d and Q/4πε₀ d² respectively with d > r as shown in Figure given with the following paragraph though its starting point is coincident with that of r. The variations of the potential and electric intensity with distance for a charged sphere are shown in Figure given with the following paragraph.

Equipotential Surfaces :

An equipotential surface is a surface in an electric field such that all points on it are at the same potential. For example, different spherical surfaces around a charged sphere are equipotential surfaces. One important property of an equipotential surface is that the direction of the electric field strength and flux density is always at right angles to the surface. Also, electric flux emerges out normal to such a surface. If, it is not so, then there would be some component of E along the surface resulting in potential difference between various points lying on it which is contrary to the definition of an equipotential surface.