Potential Energy in External Field — Revision Notes

⚡ 30-Second Revision

Single Charge: — $U = qV$
System of Two Charges: — $U = q_1 V_1 + q_2 V_2 + k \frac{q_1 q_2}{r_{12}}$
Electric Dipole (Uniform Field): — $U = -\vec{p} \cdot \vec{E} = -pE \cos\theta$
Work Done by External Agent (no $\Delta K$): — $W_{ext} = \Delta U = U_f - U_i$
Work Done by Electric Field: — $W_{field} = -\Delta U = U_i - U_f$
Stable Equilibrium (Dipole): — $\theta = 0^\circ$, $U_{min} = -pE$
Unstable Equilibrium (Dipole): — $\theta = 180^\circ$, $U_{max} = +pE$
Reference Point: — $U=0$ at infinity (for charges), $U=0$ at $\theta=90^\circ$ (for dipoles).

2-Minute Revision

Potential energy in an external electric field quantifies the energy stored when charges or dipoles are placed in a pre-existing electric environment. For a single point charge $q$ at a location with external potential $V$, the potential energy is $U=qV$.

This is the work done to bring the charge from infinity to that point. For a system of multiple charges, the total potential energy is the sum of each charge's potential energy in the external field plus the interaction potential energy between all pairs of charges.

For example, two charges $q_1, q_2$ at $V_1, V_2$ with separation $r_{12}$ have $U = q_1 V_1 + q_2 V_2 + k \frac{q_1 q_2}{r_{12}}$. An electric dipole with moment $\vec{p}$ in a uniform external electric field $\vec{E}$ has potential energy $U = -\vec{p} \cdot \vec{E} = -pE \cos\theta$.

This energy is minimum (stable equilibrium) when the dipole aligns with the field ($\theta=0^\circ$) and maximum (unstable equilibrium) when anti-aligned ($\theta=180^\circ$). The work done by an external agent to change a configuration without kinetic energy change equals the change in potential energy, $W_{ext} = \Delta U$.

Always pay attention to signs and units.

5-Minute Revision

The concept of potential energy in an external electric field is crucial for understanding how charged particles and dipoles behave in pre-existing electric environments. It's fundamentally linked to the work done by conservative electrostatic forces.

For a single point charge $q$ placed at a point where the external electric potential is $V$, its potential energy is directly given by $U = qV$. This value represents the work an external agent must do to bring the charge from a reference point (usually infinity, where $V=0$) to its current position without accelerating it.

Remember that potential $V$ is energy per unit charge, while potential energy $U$ is the total energy of the charge.

When dealing with a system of multiple charges, say $q_1$ and $q_2$ at positions $\vec{r_1}$ and $\vec{r_2}$ respectively, in an external field, the total potential energy is the sum of three components: the potential energy of $q_1$ in the external field ($q_1 V(\vec{r_1})$), the potential energy of $q_2$ in the external field ($q_2 V(\vec{r_2})$), and the interaction potential energy between $q_1$ and $q_2$ ($k \frac{q_1 q_2}{r_{12}}$).

So, $U_{total} = q_1 V(\vec{r_1}) + q_2 V(\vec{r_2}) + k \frac{q_1 q_2}{r_{12}}$. This formula can be extended for more charges by summing all individual $q_i V_i$ terms and all pairwise interaction terms.

For an electric dipole with dipole moment $\vec{p}$ in a uniform external electric field $\vec{E}$, its potential energy is given by the scalar product $U = -\vec{p} \cdot \vec{E}$. This can also be written as $U = -pE \cos\theta$, where $\theta$ is the angle between the dipole moment vector and the electric field vector.

The potential energy is minimum (most stable equilibrium) when $\theta = 0^\circ$ ($U = -pE$), meaning the dipole is aligned with the field. It is maximum (unstable equilibrium) when $\theta = 180^\circ$ ($U = +pE$), meaning the dipole is anti-aligned.

The work done by an external agent to rotate a dipole from an initial angle $\theta_i$ to a final angle $\theta_f$ is $W_{ext} = U_f - U_i = -pE(\cos\theta_f - \cos\theta_i)$.

Key points for NEET: Always pay close attention to the signs of charges and potentials. Convert all units to SI. Distinguish between work done by the external agent (which increases potential energy) and work done by the electric field (which decreases potential energy). Remember that for a system with zero net charge in a uniform external potential, the potential energy contribution from the external field is zero, but the internal interaction energy might still be non-zero.

Prelims Revision Notes

1
Electric Potential Energy (U): — Energy stored in a charge configuration due to its position in an electric field. Scalar quantity, SI unit: Joule (J).
2
Potential Energy of a Single Point Charge: — For a charge $q$ at a point where external potential is $V$, $U = qV$. Reference point for $V$ is usually infinity ($V=0$).
3
Potential Energy of a System of Charges in an External Field:

* For two charges $q_1, q_2$ at $\vec{r_1}, \vec{r_2}$ in external potentials $V(\vec{r_1}), V(\vec{r_2})$: $U = q_1 V(\vec{r_1}) + q_2 V(\vec{r_2}) + \frac{1}{4piepsilon_0} \frac{q_1 q_2}{r_{12}}$ * Generalization for multiple charges: Sum of $q_i V_i$ for each charge, plus sum of $k \frac{q_i q_j}{r_{ij}}$ for all unique pairs.

1
Potential Energy of an Electric Dipole in a Uniform External Electric Field:

* $U = -\vec{p} \cdot \vec{E} = -pE \cos\theta$, where $\theta$ is the angle between dipole moment $\vec{p}$ and electric field $\vec{E}$. * Stable Equilibrium: $\theta = 0^\circ$, $U_{min} = -pE$. Dipole aligns with field. * Unstable Equilibrium: $\theta = 180^\circ$, $U_{max} = +pE$. Dipole anti-aligns with field. * Reference Point: Often $U=0$ when $\theta = 90^\circ$.

1
Work Done by External Agent: — To move a charge/dipole from initial state $i$ to final state $f$ without changing kinetic energy:

* $W_{ext} = \Delta U = U_f - U_i$

1
Work Done by Electric Field: — $W_{field} = -\Delta U = U_i - U_f$
2
Non-uniform Electric Field: — An electric dipole in a non-uniform field experiences both a net force and a net torque. In a uniform field, it experiences only torque.
3
Key Considerations:

* Signs: Crucial for charges ($+q$ or $-q$) and potentials ($+V$ or $-V$). * Units: Always use SI units (Coulombs, Volts, meters, Joules). * Zero Net Charge in Uniform Potential: If $\sum q_i = 0$ and the external potential is uniform, the potential energy contribution from the external field ($\sum q_i V_i$) is zero. However, interaction energy still exists.

Vyyuha Quick Recall

PE = qV, Dipole = -pE cos(theta) -> 'PE is qV, Dipole's PE is Negative PE Cost (of theta)'