Relation between field and potential Derivation

Electric Potential Due to a Group of Charges

    We know that electric potential is a scalar quantity that represents the potential energy per unit charge at a point in space due to a source charge or a group of charges. 

    Consider a point P where we want to find out electric potential due to a group of charges `q_1, q_2,\ q_.............,q_n`. These charges are located at distances  `r_1, r_2,\ r_3.............,r_n` from point P.

Electric Potential Due to a Group of Charges

    Electric potential due to a group of charges is the scalar sum of the potentials from each charge.

`V  =\ V_1+\ V_2\ +\ V_3\ +\ \ldots\ldots\ldots\ldots..+V_n`

`V\ =\ \frac{k\ q_1}{r_1}+\ \frac{k\ q_2}{r_2}\ +\ \frac{k\ q_3}{r_3}\ +\ \ldots\ldots\ldots\ldots..+\frac{k\ q_n}{r_n}`

Relation between Electric Field and Electric Potential

Electric field and electric potential are closely related concepts in electromagnetism, and they are mathematically linked.

Electric Field `(\vec{E})`

The electric field is a vector quantity that represents the force experienced by a positive test charge at a point in space. It points in the direction of the force that a positive charge could experience.

Its units are newtons per coulomb (N/C) or volts per meter (V/m).

Electric Potential (V) 

The electric potential is a scalar quantity that represents the electric potential energy per unit charge at a point in space.

Its units are volts (V), where 1 volt = 1 joule/coulomb (J/C).

The difference in electric potential between two points is called the ‘Potential Difference’ or ‘Voltage’.

Derivation of Relation between Electric Field and Electric Potential

Consider two equipotential surfaces with potential V (at high potential) and V – dV (at low potential). The direction of the electric field is from the surface with high potential to the surface with low potential.

Work is done when a unit charge is moved from point B to point A in this electric field.

`W_{BA}\ =\vec{F}.\vec{dr}` 

`W_{BA}\ =\vec{E}.\vec{dr} ` 

`W_{BA}\ =E\ dr\ cos\ 180^o`

`W_{BA}\ =E\ dr\ (\ -\ 1)`

`W_{BA}\ =-\ E\ dr\ `                    ……  eq (1)

    The work done (W) to move a charge (q) between two points with different potentials is given by 

`V_A\ -\ V_B\ =\ \frac{W_{BA}}{q}`

`(V)\ -\ (V\ -\ dV)\ =\ \frac{W_{BA}}{1}`

`V\ \ -\ \ V\ +\ dV\ =\ W_{BA}`

`dV\ =\ W_{BA}`

`\ W_{BA}\ =\ dV `         ……  eq (2)

By equations 1 and 2

`-\ E\ dr\ =\ dV `

`E\ =\ -\ \frac{dV}{dr}`

This is the relation between the Electric field and electric potential.

NCERT Chapter 2 Physics class 12

• Electric Potential
• Potential due to a Point Charge
• Electric Potential due to Charged Conducting Sphere with Derivation
• Electric Potential Due to Charged Conducting Sphere
• Electric Potential Due to Charged Non-Conducting Sphere
• Electric Potential due to Dipole at any Point
• Equipotential Surfaces and Properties of Equipotential Surface
• Electric Potential Due to Electric Dipole
• Electric Potential Due to a Group of Charges and Relation between Electric Field and Potential
• Electric Potential Energy of a System of Two-Point Charges
• Define the Electrostatic Potential Energy of a System of Two and Three-Point Charges
• Work in Rotation of Electric Dipole in Electric Field
• Potential Energy of a Dipole in an External Field
• Electrostatics of Conductors
• Dielectrics and polarization