Electric Potential — Revision Notes

⚡ 30-Second Revision

Electric Potential (V): — Work done per unit charge from infinity to a point. Scalar quantity. Unit: Volt (V) or J/C.
Potential due to Point Charge: — $V = \frac{1}{4\pi\epsilon_0} \frac{Q}{r} = \frac{kQ}{r}$.
Potential due to System of Charges: — $V_{total} = \sum V_i = \sum \frac{kQ_i}{r_i}$ (algebraic sum).
Potential Difference: — $\Delta V = V_B - V_A = \frac{W_{ext}}{q_0}$.
Work Done: — $W = q(V_B - V_A)$.
Electric Potential Energy (U): — Energy of a charge $q$ at potential $V$ is $U = qV$. Unit: Joule (J).
Potential Energy of Two Charges: — $U = \frac{1}{4\pi\epsilon_0} \frac{q_1 q_2}{r_{12}} = \frac{kq_1q_2}{r_{12}}$.
Potential Energy of Dipole in E-field: — $U = -\vec{p} \cdot \vec{E} = -pE \cos\theta$.
Relation between E and V: — $\vec{E} = -\nabla V$ (or $E_x = -\frac{dV}{dx}$ in 1D).
Equipotential Surfaces: — Surfaces of constant potential. Electric field lines are perpendicular to them. No work done along them.

2-Minute Revision

Electric potential is a scalar measure of the electric field's influence at a point, defined as the work required to bring a unit positive charge from infinity to that point. It's measured in Volts (J/C).

For a point charge $Q$, potential at distance $r$ is $V = kQ/r$. For multiple charges, sum their potentials algebraically. Potential difference, $\Delta V$, is the work done per unit charge between two points.

The work done to move a charge $q$ through a potential difference $\Delta V$ is $W = q\Delta V$. Electric potential energy ($U$) is the energy of a charge $q$ at potential $V$, given by $U=qV$. For a system of charges, $U$ is the sum of potential energies of all unique pairs ($U = \sum k q_i q_j / r_{ij}$).

An electric dipole $\vec{p}$ in an external field $\vec{E}$ has potential energy $U = -\vec{p} \cdot \vec{E}$. Crucially, the electric field $\vec{E}$ is the negative gradient of the potential, $\vec{E} = -\nabla V$, meaning field lines point towards decreasing potential and are perpendicular to equipotential surfaces (surfaces of constant potential).

Remember that zero potential does not necessarily mean zero electric field, and vice-versa.

5-Minute Revision

Electric potential, $V$, is a scalar quantity representing the potential energy per unit positive charge at a given point in an electric field. It's defined as the work done by an external agent to bring a unit positive test charge from infinity to that point without acceleration.

The SI unit is the Volt (V), equivalent to Joules per Coulomb (J/C). The potential due to a single point charge $Q$ at a distance $r$ is $V = \frac{1}{4\pi\epsilon_0} \frac{Q}{r}$. For a system of multiple point charges, the total potential at any point is the algebraic sum of the potentials due to each individual charge: $V_{total} = \sum_{i} \frac{kQ_i}{r_i}$.

Electric potential difference, $\Delta V = V_B - V_A$, is the work done per unit charge to move a test charge from point A to point B. Consequently, the work done by an external agent to move a charge $q$ from A to B is $W = q(V_B - V_A)$.

Electric potential energy, $U$, is the energy stored in a charge configuration. For a single charge $q$ at a point with potential $V$, its potential energy is $U = qV$. For a system of two charges $q_1$ and $q_2$ separated by a distance $r_{12}$, their mutual potential energy is $U = \frac{1}{4\pi\epsilon_0} \frac{q_1 q_2}{r_{12}}$.

For a system of $n$ charges, the total potential energy is the sum of the potential energies of all unique pairs, $U = \sum_{i<j} \frac{k q_i q_j}{r_{ij}}$. An electric dipole with moment $\vec{p}$ in a uniform external electric field $\vec{E}$ has potential energy $U = -\vec{p} \cdot \vec{E} = -pE \cos\theta$, where $\theta$ is the angle between $\vec{p}$ and $\vec{E}$.

Minimum potential energy occurs at $\theta = 0^{\circ}$ (stable equilibrium), and maximum at $\theta = 180^{\circ}$ (unstable equilibrium).

Equipotential surfaces are surfaces where the electric potential is constant. Electric field lines are always perpendicular to these surfaces, and no work is done in moving a charge along an equipotential surface.

The electric field $\vec{E}$ is related to the electric potential $V$ by $\vec{E} = -\nabla V$. This means the electric field points in the direction of the steepest decrease in potential. If potential is constant in a region, the electric field is zero there.

Conversely, zero potential does not always mean zero field (e.g., equatorial plane of a dipole).

Prelims Revision Notes

Electric Potential (V)

Definition: — Work done by external agent to bring unit positive test charge from infinity to a point without acceleration. $V = W/q_0$.
Unit: — Volt (V) or J/C. Scalar quantity.
Potential due to Point Charge Q: — $V = \frac{1}{4\pi\epsilon_0} \frac{Q}{r} = \frac{kQ}{r}$. (Remember sign of Q).
Potential due to System of Point Charges: — $V_{total} = \sum_{i} V_i = \sum_{i} \frac{kQ_i}{r_i}$ (algebraic sum).
Potential Difference ($\Delta V$): — $V_B - V_A = \frac{W_{ext}(A \to B)}{q_0}$.
Work Done by External Agent: — $W_{ext} = q(V_B - V_A)$.

Electric Potential Energy (U)

Definition: — Energy stored in a system of charges due to their configuration. Unit: Joule (J).
Of a Charge q in Potential V: — $U = qV$.
Of Two Point Charges ($q_1, q_2$) separated by $r_{12}$: — $U = \frac{1}{4\pi\epsilon_0} \frac{q_1 q_2}{r_{12}} = \frac{kq_1q_2}{r_{12}}$. (Remember signs of $q_1, q_2$).
Of a System of Multiple Charges: — $U_{total} = \sum_{i<j} \frac{kq_i q_j}{r_{ij}}$ (sum over all unique pairs).
Of an Electric Dipole ($\vec{p}$) in Uniform Electric Field ($\vec{E}$): — $U = -\vec{p} \cdot \vec{E} = -pE \cos\theta$.

* Stable equilibrium: $\theta = 0^{\circ}$, $U = -pE$ (minimum). * Unstable equilibrium: $\theta = 180^{\circ}$, $U = +pE$ (maximum). * Perpendicular: $\theta = 90^{\circ}$, $U = 0$.

Relation between Electric Field ($\vec{E}$) and Electric Potential (V)

$\vec{E} = -\nabla V$. In 1D: $E_x = -\frac{dV}{dx}$.
Electric field points in the direction of decreasing potential.
Magnitude of E is the rate of change of V with distance.

Equipotential Surfaces

Definition: — Surfaces where electric potential is constant.
Properties:

* Electric field lines are always perpendicular to equipotential surfaces. * No work is done in moving a charge along an equipotential surface. * They never intersect each other. * Closer spacing indicates a stronger electric field.

Examples: — Concentric spheres for a point charge; parallel planes for a uniform electric field.

Key Concepts to Differentiate

Potential vs. Potential Energy: — $V$ is per unit charge, $U$ is for a specific charge ($U=qV$).
Zero E-field vs. Zero Potential: — Not equivalent. Inside a conductor, $E=0$ but $V \ne 0$. On dipole's equatorial plane, $V=0$ but $E \ne 0$.

Vyyuha Quick Recall

To remember the relationship between Electric Field and Potential: Electric field Decreases Voltage. Think of 'EDV' as 'Electric Drives Voltage'. The negative sign implies the field points in the direction of decreasing potential.