19.1 Electric Potential Energy and Electric Potential | Electric Potential And Capacitor | Physics 12

Electric Potential Energy ( $U$ )

Electric potential energy is the energy a charge possesses by virtue of its position in an electric field. It is defined as the work done by an external agent in bringing a positive test charge $q_{0}$ from infinity (the reference point of zero potential energy) to a given point in the field, without any acceleration:

$U = W_{external} = - W_{field}$

The SI unit of electric potential energy is the Joule (J).

Key Points

When a positive charge moves against the electric field (from lower to higher potential), an external agent does positive work, and the electric potential energy increases.
When a positive charge moves along the electric field (from higher to lower potential), the field does positive work, and the electric potential energy decreases.

Electric Potential ( $V$ )

Electric potential at a point in an electric field is defined as the electric potential energy per unit positive test charge placed at that point:

$V = \frac{U}{q _{0}} = \frac{W}{q _{0}}$

where $W$ is the work done in bringing charge $q_{0}$ from infinity to that point.

SI Unit: Volt (V)

One Volt is defined as the potential at a point if one Joule of work is done in bringing a positive charge of one Coulomb from infinity to that point:

$1 V = 1 J C^{- 1} = 1 N m C^{- 1}$

Potential Difference

The potential difference $Δ V$ between two points A and B is:

$Δ V = V_{A} - V_{B} = \frac{W _{A \to B}}{q _{0}}$

This represents the work done per unit charge in moving a positive test charge from B to A.

Direction of Motion of Charges

A positive charge naturally moves from a region of higher potential to lower potential (in the direction of the electric field).
A negative charge naturally moves from a region of lower potential to higher potential (opposite to the electric field).
Moving a positive charge against the field requires work by an external agent, increasing its potential energy.

Relationship Between Electric Field and Potential Gradient

The electric field intensity $E$ is related to the electric potential $V$ by:

$E = - \frac{Δ V}{Δ r}$

The quantity $\frac{Δ V}{Δ r}$ is called the potential gradient. The negative sign indicates that the electric field points in the direction of decreasing potential (from high $V$ to low $V$ ).

The SI unit of electric field can therefore also be expressed as V m⁻¹ (equivalent to N C⁻¹).

Electron-Volt (eV)

A convenient unit of energy in atomic and nuclear physics is the electron-volt (eV):

One electron-volt is the kinetic energy gained by an electron when it is accelerated through a potential difference of 1 Volt.

$1 eV = 1.6 \times 1 0^{- 19} J$

In general, a charge $q$ accelerated through potential difference $V$ gains kinetic energy:

$Δ K E = q V$

For an electron accelerated through $1000 V$ : $Δ K E = 1000 eV$ .

Summary Table

Quantity	Symbol	Formula	SI Unit
Electric Potential Energy	$U$	$U = q_{0} V$	Joule (J)
Electric Potential	$V$	$V = W / q_{0}$	Volt (V) = J C⁻¹
Potential Gradient	$Δ V /Δ r$	—	V m⁻¹
Electric Field	$E$	$E = - Δ V /Δ r$	V m⁻¹ or N C⁻¹

Electric Potential Energy ( $U$ )

$U = W_{external} = - W_{field}$

The SI unit of electric potential energy is the Joule (J).

Key Points

When a positive charge moves against the electric field (from lower to higher potential), an external agent does positive work, and the electric potential energy increases.
When a positive charge moves along the electric field (from higher to lower potential), the field does positive work, and the electric potential energy decreases.

Electric Potential ( $V$ )

Electric potential at a point in an electric field is defined as the electric potential energy per unit positive test charge placed at that point:

$V = \frac{U}{q _{0}} = \frac{W}{q _{0}}$

where $W$ is the work done in bringing charge $q_{0}$ from infinity to that point.

SI Unit: Volt (V)

One Volt is defined as the potential at a point if one Joule of work is done in bringing a positive charge of one Coulomb from infinity to that point:

$1 V = 1 J C^{- 1} = 1 N m C^{- 1}$

Potential Difference

The potential difference $Δ V$ between two points A and B is:

$Δ V = V_{A} - V_{B} = \frac{W _{A \to B}}{q _{0}}$

This represents the work done per unit charge in moving a positive test charge from B to A.

Direction of Motion of Charges

A positive charge naturally moves from a region of higher potential to lower potential (in the direction of the electric field).
A negative charge naturally moves from a region of lower potential to higher potential (opposite to the electric field).
Moving a positive charge against the field requires work by an external agent, increasing its potential energy.

Relationship Between Electric Field and Potential Gradient

The electric field intensity $E$ is related to the electric potential $V$ by:

$E = - \frac{Δ V}{Δ r}$

The SI unit of electric field can therefore also be expressed as V m⁻¹ (equivalent to N C⁻¹).

Electron-Volt (eV)

A convenient unit of energy in atomic and nuclear physics is the electron-volt (eV):

One electron-volt is the kinetic energy gained by an electron when it is accelerated through a potential difference of 1 Volt.

$1 eV = 1.6 \times 1 0^{- 19} J$

In general, a charge $q$ accelerated through potential difference $V$ gains kinetic energy:

$Δ K E = q V$

For an electron accelerated through $1000 V$ : $Δ K E = 1000 eV$ .

Summary Table

Quantity	Symbol	Formula	SI Unit
Electric Potential Energy	$U$	$U = q_{0} V$	Joule (J)
Electric Potential	$V$	$V = W / q_{0}$	Volt (V) = J C⁻¹
Potential Gradient	$Δ V /Δ r$	—	V m⁻¹
Electric Field	$E$	$E = - Δ V /Δ r$	V m⁻¹ or N C⁻¹