Comprehensive Notes on Electric Potential and Dipoles

Electric Potential

• Definition of Electric Potential ($V$): The electric potential at a point inside an electric field is defined as the amount of work done by an external agent to bring a unit positive charge from infinity to that specific point without changing its Kinetic Energy ($KE$).

• Alternative Definition: It is also described as the work done per unit charge ($\frac{W}{q}$).

• Nature of Quantity: Electric potential is a scalar quantity.

• S.I. Unit: The standard unit is the Volt ($V$), which is equivalent to Joules per Coulomb ($J/C$).

• Dimensional Formula: The dimension is given as $[M^1 L^2 T^{-3} A^{-1}]$.

Work Done by a Rotating Dipole in a Uniform Electric Field

• To calculate the work done to rotate a dipole within a uniform electric field ($E$), we begin with the expression for torque ($\tau$) acting on a dipole:   - $\tau = pE \sin(\theta)$

• Small Work Done: The work done ($dW$) to rotate the dipole through a very small angle ($d\theta$) is expressed as:   - $dW = \tau \, d\theta$   - Substituting the torque expression: $dW = pE \sin(\theta) \, d\theta$

• Total Work Done: To find the total work done ($W$) to rotate the dipole from an initial angle $\theta_1$ to a final angle $\theta_2$, we integrate the expression:   - $W = \int_{\theta_1}^{\theta_2} pE \sin(\theta) \, d\theta$   - $W = pE \int_{\theta_1}^{\theta_2} \sin(\theta) \, d\theta$   - $W = pE [-\cos(\theta)]_{\theta_1}^{\theta_2}$   - $W = -pE (\cos(\theta_2) - \cos(\theta_1))$   - Re-arranging for the standard form: $W = pE (\cos(\theta_1) - \cos(\theta_2))$

• Specific Case (Displacement from Field Direction):   - If the dipole is displaced starting from the direction of the field (where $\theta_1 = 0^{\circ}$) to a general angle $\theta_2 = \theta$, the formula becomes:   - $W = pE (\cos(0^{\circ}) - \cos(\theta))$   - $W = pE (1 - \cos(\theta))$   - This work done is stored in the dipole in the form of Potential Energy ($U$).

Relation Between Electric Field and Potential

• The general relation between the Electric Field ($E$) and Electric Potential ($V$) is derived from the work done moving a charge between two points (A to B):   - $dW = \mathbf{F}_{ext} \cdot d\mathbf{r}$   - Since $\mathbf{F}_{ele} = q\mathbf{E}$, then $\mathbf{F}_{ext} = -q\mathbf{E}$.   - $dW = -q\mathbf{E} \cdot d\mathbf{r}$   - $V_B - V_A = \frac{dW}{q} = -\int \mathbf{E} \cdot d\mathbf{r}$

• Differential Form: $dv = -E \, dr \cos(\theta)$

• Potential Gradient: If we move along the direction of the field ($\theta = 0^{\circ}$), then:   - $dv = -E \, dr$   - $E = -\frac{dv}{dr}$   - Here, $\frac{dv}{dr}$ is called the potential gradient. The negative sign indicates that in the direction of the electric field, the potential value decreases.

• Case of Perpendicular Movement:   - When moving perpendicular to the direction of the field ($\theta = 90^{\circ}$):   - $dv = -E \, dr \cos(90^{\circ})$   - Since $\cos(90^{\circ}) = 0$, $dv = 0$, which means $V = \text{constant}$.   - This indicates that when moving perpendicular to the field, the potential remains constant.

• Units of Electric Field: Based on this relation, the unit for Electric Field can also be expressed as $V/m$ (Volts per meter) or $N/C$ (Newtons per Coulomb).

Potential Difference and Distance

• The potential difference between two points A and B situated at distances $r_A$ and $r_B$ from a charge $q$ is:   - $V_B - V_A = \frac{q}{4\pi \epsilon_0} \left( \frac{1}{r_B} - \frac{1}{r_A} \right)$

• Observations on Movement:   - If a charge moves opposite to the direction of the field (r_B < r_A), then the potential will increase (V_B > V_A).

Electric Potential Energy ($U$)

• Definition: The amount of work done by an external agent to bring a system of charges together from infinity without changing their kinetic energy.

• Potential Energy of a Dipole in a Uniform Electric Field:   - To calculate this, we use the work done to rotate a dipole from an initial reference position to a final position.   - Reference position ($\theta_1$): Usually taken as $90^{\circ}$ (the position of zero potential energy).   - Final position ($\theta_2$): $\theta$   - $W = pE (\cos(90^{\circ}) - \cos(\theta))$   - $W = pE (0 - \cos(\theta))$   - $W = -pE \cos(\theta)$   - This work done is stored as the potential energy of the dipole: $U = -\mathbf{p} \cdot \mathbf{E}$.

Maximum and Minimum Potential Energy

• Maximum Potential Energy (Unstable Equilibrium):   - Occurs when $\theta = 180^{\circ}$.   - $U = -pE \cos(180^{\circ}) = -pE(-1)$   - $U_{max} = pE$

• Minimum Potential Energy (Stable Equilibrium):   - Occurs when $\theta = 0^{\circ}$.   - $U = -pE \cos(0^{\circ}) = -pE(1)$   - $U_{min} = -pE$

Force on a Dipole in a Non-Uniform Electric Field

• The relationship between conservative force and potential energy is given by:   - $F = -\frac{dU}{dr}$

• Substituting the potential energy of the dipole $U = -pE$:   - $F = -\frac{d}{dr}(-pE)$   - $F = p \frac{dE}{dr}$

Equipotential Surface

• Definition: A surface where the electric potential value is the same at every point due to a charge configuration.

• Properties:   1. Potential Difference: The potential difference ($V_B - V_A$) between any two points on an equipotential surface is zero ($0$).   2. Work Done: The work done to move a charge from one point to another on an equipotential surface is zero ($W_{AB} = q(V_B - V_A) = 0$).   3. Perpendicularity to Field: Equipotential surfaces are always perpendicular to the electric field lines.      - Proof: For the surface, $dv = 0$. Since $dv = -\mathbf{E} \cdot d\mathbf{r} = -E \, dr \cos(\theta)$, for $dv$ to be zero, $\cos(\theta)$ must be zero, implying $\theta = 90^{\circ}$.   4. Geometric Shape: For a point charge, the equipotential surfaces are spherical.   5. No Intersection: Two equipotential surfaces can never intersect each other. If they did, there would be two different directions of electric intensity at the point of intersection, which is physically impossible.

Electric Potential Due to an Arbitrary Point Near a Dipole

• To find the net potential at a point at distance $r$ from the center of the dipole making an angle $\theta$ with the dipole axis:

• Potential due to $-q$ charge:   - Let $AC$ be the distance from $-q$ to the point.   - Using geometry (drawing perpendicular $AN$), $AC \approx r + l \cos(\theta)$.   - $V_{(-)} = \frac{1}{4\pi \epsilon_0} \frac{-q}{(r + l \cos(\theta))}$

• Potential due to $+q$ charge:   - Let $BC$ be the distance from $+q$ to the point.   - Using geometry (drawing perpendicular $BM$), $BC \approx r - l \cos(\theta)$.   - $V_{(+)} = \frac{1}{4\pi \epsilon_0} \frac{q}{(r - l \cos(\theta))}$

• Net Potential ($V$):   - $V = V_{(+)} + V_{(-)}$   - $V = \frac{q}{4\pi \epsilon_0} \left( \frac{1}{r - l \cos(\theta)} - \frac{1}{r + l \cos(\theta)} \right)$   - $V = \frac{q}{4\pi \epsilon_0} \left( \frac{(r + l \cos(\theta)) - (r - l \cos(\theta))}{r^2 - l^2 \cos^2(\theta)} \right)$   - $V = \frac{q(2l \cos(\theta))}{4\pi \epsilon_0 (r^2 - l^2 \cos^2(\theta))}$   - Since $p = q \times 2l$, the general formula is: $V = \frac{p \cos(\theta)}{4\pi \epsilon_0 (r^2 - l^2 \cos^2(\theta))}$

Electric Potential Due to a Short Dipole on the Axial Line

• Condition for Short Dipole: $l \ll r$.

• Case I: The point $C$ is on the axial line (End-on position).

• Distance Calculation:   - Distance from $+q$ to point $C$ is $BC = r - l$.   - Distance from $-q$ to point $C$ is $AC = r + l$.

• Potentials:   - $V_{(+)} = \frac{1}{4\pi \epsilon_0} \frac{q}{(r - l)}$   - $V_{(-)} = \frac{1}{4\pi \epsilon_0} \frac{-q}{(r + l)}$

• Net Potential at Axial Point:   - $V = V_{(+)} + V_{(-)}$   - $V = \frac{1}{4\pi \epsilon_0} \left( \frac{q}{r - l} - \frac{q}{r + l} \right)$   - $V = \frac{q}{4\pi \epsilon_0} \left( \frac{r + l - (r - l)}{r^2 - l^2} \right)$   - $V = \frac{q(2l)}{4\pi \epsilon_0 (r^2 - l^2)}$   - Substituting $p = q \times 2l$ and assuming $r^2 - l^2 \approx r^2$ for a short dipole:   - $V = \frac{p}{4\pi \epsilon_0 r^2}$