Electric Potential Due to a Point Charge Derivation

The energy stored in the object that depends upon the position of various parts of that object or system is the potential energy of that system. In simpler words, it is the energy possessed by a body at rest and the actual potential energy of an object depends on its position relative to other objects. Electric potential energy is the amount of negative work done by an electric force to bring a given object from one point to the other desired point. The result obtained by the division of this electric potential energy with the quantity of charge is the electric potential. Point charges are the fundamental building blocks of matter (for example electrons). Let’s study the potential energy and electric potential due to point charge in brief. 

Potential Energy

The potential energy of any object refers to the energy stored within it (the body is at rest). This energy remains stored due to its configuration such as position or orientation. For example, the gravitational potential energy for a hot air balloon depends upon the position within the Earth’s gravitational field, which means it will change depending upon how far or near it is from the Earth’s centre. Thus, we can say that the actual potential energy of an object depends on its position relative to other objects.

Two Factors on Which Potential Energy Depends

The two factors on which the potential energy depends are as follows:

• The first factor is the mass of the body. The mass of the body is directly proportional to the potential energy of the body. It means the more mass the body possesses, the greater will be its potential energy.
• The second factor shall be the height above the ground. It is also directly proportional to the potential energy, and on increasing the height, the potential energy increases too.

What is Electric Potential?

As already discussed, the electric potential is the quotient obtained on the division of electric potential energy with the quantity of charge is the electric potential. Moreover, the strength of any electric field will depend upon this electric potential and does not depend on the condition that whether the charge is present in the electric field or not. 

There are two basic factors on which this electric potential depends, which are as follows:

• The electric charge is carried by the object.
• The relative position of the given object with respect to all other charged objects present.

Electric Potential due to Point Charge Derivation

The electric potential at a given point is the amount of work needed for carrying a given unit positive charge from infinity to the given point against the present electric field.

Let’s have a look at the Electric Potential due to Point Charge Derivation

The expression of work done for moving a given unit charge is as follows:

∆W=Fdx

∆W =F dx cos180

∆W = -F dx

=Q4πϵ0x2dx1x

The expression for total work done is as follows:

W= r14πϵ0 Q/x2. dx 

W=-Q4πϵ0x2r1x2.dx

W=-Q4πϵ0 [ 1x] r

W=-Q4πϵ0[1/r-1/∞]

W=Q4πϵ0r

As per the definition, the work done will be equal to the electric potential (V) at a given point, therefore, V=W.

V=Q4πϵ0r

Or, V=14πϵ0.Qr

Solved Questions on Electric Potential Due To a Point Charge

Q 1- Determine the potential of a charge of 2pC at a distance of 1m due to the given charge?

Solution:

The formula for evaluating potential due to point charge is as follows:

V=14πϵ0.Qr

V=9× 109 x 2 x 10-12/1

V=9× 109 x 2 x 10-12

V=18×103

Answer: The potential of a charge of 2pC at a distance of 1m due to the given charge is 18×103.

Q 2- Determine the potential of a charge of 10pC at a distance of 0.5 m due to the charge.

Solution:

The formula for evaluating potential due to point charge is as follows:

V=14πϵ0.Qr

V=9× 109 x 10 x 10-12/0.5

V=9× 109 x 2 x 10-11

V=18×10-11

Answer: The potential of a charge of 10pC at a distance of 0.5 m due to the charge is 18×10-11.

Conclusion 

Potential energy is the energy stored in an object when the object is at rest. Moreover, this actual potential energy of an object depends on its position relative to other objects and it depends upon two factors; the mass of the body and height above the ground. The electric potential energy refers to the work done by an object to bring a unit charge from one point to another point in an electric field. Electric potential is obtained on the division of this electric potential energy with the quantity of charge. There are two basic factors on which this electric potential depends: the electric charge carried by the object and the relative position of the given object with respect to all other charged objects present.