Let’s talk about Potential Energy

1. Electric Potential Energy (Concept)

Electric potential energy (U) of a charge is the energy it possesses due to its position in an electric field.

👉 Defined using work done by an external agent.

If a charge q is moved slowly (no acceleration) in an electric field:

$$\mathrm{\Delta }U=-W_{\text{electric}}$$

Electrostatic force is conservative, so potential energy depends only on initial and final positions, not on path.

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2. Potential Energy due to a System of Charges (Internal Interaction)

(a) Two point charges

Let charges $q_1$ and $q_2$ be separated by distance $r_{12}$.

Work done to assemble the system from infinity:

$$U=\frac{1}{4\pi {\epsilon }_0}\frac{q_1{q}_2}{r_{12}}$$

✔ Positive for like charges
✔ Negative for unlike charges

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(b) System of many charges

For charges $q_1,q_2,\dots q_n$:

$$\boxed{{\displaystyle U=\frac{1}{4\pi {\epsilon }_0}\sum _{i<j}\frac{q_i{q}_j}{r_{ij}}}}$$

This is the total electrostatic potential energy of the system.

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3. Electric Potential $V$

Electric potential at a point = potential energy per unit charge

$$\boxed{{\displaystyle V=\frac{U}{q}}}$$

Unit: Volt (V)
1 V = 1 J C⁻¹

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4. Potential due to a Point Charge (Derivation)

Consider a point charge Q at origin.
Find potential at point P at distance r.

Electric field due to Q:

$$\vec{E}=\frac{1}{4\pi {\epsilon }_0}\frac{Q}{r^2}\widehat{r}$$

Work done in bringing unit positive charge from infinity to r:

$$V(r)=-{\int }_{\mathrm{\infty }}^r\vec{E}\cdot d\vec{r}$$

Since $\vec{E}\parallel d\vec{r}$:

$$V(r)=-{\int }_{\mathrm{\infty }}^r\frac{1}{4\pi {\epsilon }_0}\frac{Q}{r^2}dr$$
$$\boxed{{\displaystyle V(r)=\frac{1}{4\pi {\epsilon }_0}\frac{Q}{r}}}$$

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5. Potential due to a System of Charges (Superposition)

Potential is a scalar, so it adds algebraically.

For point P:

$$\boxed{{\displaystyle V(P)=\frac{1}{4\pi {\epsilon }_0}(\frac{q_1}{r_1}+\frac{q_2}{r_2}+\cdots )}}$$

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6. Potential due to an Electric Dipole

Dipole moment:

$$\vec{p}=q(2a)\widehat{p}$$

At point P with position vector $\vec{r}$ making angle θ with $\vec{p}$:

$$\boxed{{\displaystyle V(\vec{r})=\frac{1}{4\pi {\epsilon }_0}\frac{\vec{p}\cdot \widehat{r}}{r^2}(r\gg a)}}$$

Special cases:

• Axial line: $V=\pm \frac{1}{4\pi {\epsilon }_0}\frac{p}{r^2}$

• Equatorial plane: $V=0$

POTENTIAL ENERGY IN AN EXTERNAL FIELD

7. Potential Energy in an External Field

(a) Single charge in external field

If external potential is $V(\vec{r})$:

$$\boxed{{\displaystyle U=qV(\vec{r})}}$$

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(b) System of charges in external field

$$\boxed{{\displaystyle U=\sum q_{i}V({\vec{r}}_i)+\frac{1}{4\pi {\epsilon }_0}\sum _{i<j}\frac{q_i{q}_j}{r_{ij}}}}$$

First term → interaction with external field
Second term → mutual interaction

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(c) Electric Dipole in Uniform External Field

Dipole in uniform E, angle θ with field.

Torque:$$\vec{\tau }=\vec{p}\times \vec{E}$$

Work done in rotation:

$$\boxed{{\displaystyle U(\theta )=-\vec{p}\cdot \vec{E}=-pE\cos \theta }}$$

✔ Minimum energy when dipole aligns with field
✔ Maximum when anti-parallel

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8. Vector Electric Field and Potential $V(\vec{r})$

Potential at position vector $\vec{r}$:

$$\boxed{{\displaystyle V(\vec{r})=-{\int }_{\mathrm{\infty }}^{\vec{r}}\vec{E}\cdot d\vec{r}}}$$

This definition is general and works for any field configuration.

9. Relation between Electric Field and Potential (Very Important)

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• Consider two very closely spaced equipotential surfaces A and B.

• Potential of surface A = V

• Potential of surface B = V + dV

• The electric field E is perpendicular to equipotential surfaces.

• Distance between the two surfaces (along field direction) = dl

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Step 1: Work done (NCERT logic)

Take a unit positive test charge.

Work done in moving it from B to A against the electric field:

$$\text{Work done}=Edl$$

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Step 2: Relation with potential difference

By definition of electric potential:

$$\text{Work done}=V_A-V_B$$

But$$V_A=V,V_B=V+dV$$So,

$$V_A-V_B=V-(V+dV)=-dV$$

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Step 3: Equating both expressions

$$Edl=-dV$$

or$$\boxed{{\displaystyle E=-\frac{dV}{dl}}}$$

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10. One-Line NCERT Summary (Exam Gold)

• $U=qV$

• $V=-\int \vec{E}\cdot d\vec{r}$

• Potential is scalar, field is vector

• Equipotential surfaces ⟂ Electric field