Electrostatic Potential and Capacitance Formula Sheet

• The work done is the product of displacement and external forces causing movement of a charge $q$from point R to point P.

$$W_{RP}={\int }_R^P{\mathbf{F}}_{\text{ext}}\cdot d\mathbf{r}$$

The Work done by the external force brings a unit charge to Point A from infinity.

$$V_{A\mathrm{<br>}}=\frac{W_{A\mathrm{\to \infty }}}{q}$$

Potential Difference

$$V_{AB}=\frac{W_{AB}}{q}$$

$V_{AB}$= Potential Difference between A and B

Electric Potential Due to a Point Charge

$$V=\frac{1}{4\pi {\epsilon }_0}\cdot \frac{Q}{r}<br>\frac{}{}$$

Potential Due to a System of Charges

For a system of many charges.

$$V = \frac{1}{4 \pi \varepsilon_0} \left( \frac{Q_1}{r_1} + \frac{Q_2}{r_2} + \frac{Q_3}{r_3} + \dots \right)$$

Electric Potential Due to Electric Dipole

• At any point, making an angle $\theta$ with the dipole axis.

$$V\; =\frac{1}{4\pi {\epsilon }_0}\cdot \frac{p\cos \theta }{r^2}$$

• On the Axial Line (θ = 0°)

$$V_{\text{axial}} = \frac{1}{4 \pi \varepsilon_0} \cdot \frac{p}{r^2}$$

• ​On the Equatorial Line (θ = 90°)

$$V_{\text{equatorial}}=0$$

Equipotential Suface: Work Done

A surface where the potential at every point is equal and hence no work is required to move a charge in equipotenital surface.

W_A - W_B = 0

Relation Between Electric Field and Potential

$$|\mathbf{E}| = - \frac{\delta V}{\delta l}$$

Note: Negative of potential gradient between equipotenital surfaces ​is equal to the magnitude of electric field.

Energy is stored in a system of charges due to the positions of the charges in an electric field.

$$U = \frac{1}{4\pi \varepsilon_0} \cdot \frac{q_1 \cdot q_2}{r}$$

Electric Potential Energy Due to a System of charges

$$U = \frac{1}{4\pi \varepsilon_0} \sum_{i=1}^{n} \sum_{j=i+1}^{n} \frac{q_i q_j}{r_{ij}}$$

​Suppose there are 3 charges then:

$$U = \frac{1}{4\pi \varepsilon_0} \left( \frac{q_1 q_2}{r_{12}} + \frac{q_1 q_3}{r_{13}} + \frac{q_2 q_3}{r_{23}} \right)$$

Electric Potential Energy in External Electric Field

If a charge or a system of charges is placed in an external field:

• Potential energy of a single charge

$$U=q\cdot V$$

• Potential energy of a system of two charges

$$U = q_1 V(\mathbf{r}_1) + q_2 V(\mathbf{r}_2) + \frac{q_1 q_2}{4\pi \varepsilon_0 r_{12}}$$

r₁ and r₂ are respective positions in the field.​​

• Potential energy of a dipole in an external field

$$U=-\vec{p}\cdot \vec{E}=-pE\cos \theta$$

Dielectrics do not conduct electricity, but they can be polarized when placed in an electric field.

• Polarisation:

$$\mathbf{P}={\epsilon }_0{\chi }_e\mathbf{E}$$

A device used to store electric charge and energy in an electric field between two conducting plates. Capacitance is a measurement of charge stored per volt.

$$C=\frac{Q}{V}$$

• Capacitance of a Parallel Plate Capacitor

$$C = \varepsilon_0 \varepsilon_r \frac{A}{d}$$

• ​Capacitance of a Parallel Plate Capacitor with a Dielectric Slab (Thickness t)

$$C=\frac{{\epsilon }_{0}A}{d-t+\frac{t}{{\epsilon }_r}}$$

• Capacitors in Series

$$\frac{1}{C_{\text{eq}}}=\frac{1}{C_1}+\frac{1}{C_2}+\frac{1}{C_3}+\dots$$

• Capacitors in Parallel

$$C_{\text{eq}}=C_1+C_2+C_3+\dots$$

• When Q and V are given:

$$U = \frac{1}{2} Q V$$

• When C and V are given:

$$U = \frac{1}{2} C V^2When C and V are given:When C and V are given:$$

• When Q and C are given:

$$U=\frac{Q^2}{2C}$$