Coulomb Force Between Two Point Charges:

$$\vec{F}=\frac{1}{4 \pi \varepsilon_{0} \varepsilon_{r}} \frac{q_1 q_2}{|\vec{r}|^{3}} \vec{r}=\frac{1}{4 \pi \varepsilon_{0} \varepsilon_{r}} \frac{q_1 q_2}{|\vec{r}|^{2}} \hat{r}$$

• The electric field intensity at any point is the force experienced by unit positive charge, given by $$\vec{E}=\frac{\vec{F}}{q_{0}}$$

• Electric force on a charge ’ $q$ ’ at the position of electric field intensity $\vec{E}$ produced by some source charges is: $$\vec{F}=q \vec{E}$$

Electric Potential:

If $\left(W _{\infty}\right) _{\text {ext }}$ is the work required in moving a point charge $q$ from infinity to a point P, the electric potential of the point P is

$$\left.V _p=\frac{\left(W _{\infty p}\right) _{e x t}}{q}\right| _{a c c=0}$$

Potential Difference Between Points A and B:

$$ V_{B} -V_{A}=-\int_{A}^{B} \vec{E} \cdot d \vec{r} $$

$$\vec{E} = -\left[\hat{i} \frac{\partial}{\partial x} V+\hat{j} \frac{\partial}{\partial x} V+\hat{k} \frac{\partial}{\partial z} V\right]$$

$$= -\left[\hat{i} \frac{\partial}{\partial x}+\hat{j} \frac{\partial}{\partial x}+\hat{k} \frac{\partial}{\partial z}\right] $$

$$=- \nabla V = -grad V$$

Formulae of $\vec{E}$ and potential V

(i) Point charge:

$$E=\frac{K q}{|\vec{r}|^{2}} \cdot \hat{r}=\frac{K q}{r^{3}} \vec{r}$$

$$V=\frac{K q}{r}$$

(ii) Infinitely long line charge:

$$\frac{\lambda}{2 \pi \varepsilon_{0} r} \hat{r}=\frac{2 K \lambda \hat{r}}{r}$$

$$ \text{V= not defined}, V_{B}-V_{A}=-2 ~K \lambda \ln \left(r_{B} / r_{A}\right)$$

(iii) Infinite non-conducting thin sheet:

$$\frac{\sigma}{2 \varepsilon_{0}} \hat{n}$$

$$\text{V= not defined}, V_{B}-V_{A}=-\frac{\sigma}{2 \varepsilon_{0}}\left(r_{B}-r_{A}\right)$$

(iv) Uniformly charged ring

$$E_{\text {axis }}=\frac{KQx}{\left(R^{2}+x^{2}\right)^{3 / 2}}, \quad E_{\text {centre }}=0$$

$$V_{\text {axis }}=\frac{KQ}{\sqrt{R^{2}+x^{2}}}, \quad ~V_{\text {centre }}=\frac{KQ}{R}$$

Where: $x$ is the distance from centre along axis.

(v) Infinitely large charged conducting sheet:

$$\frac{\sigma}{\varepsilon_{0}} \hat{n}$$

$$\text{V= not defined}, V_{B}-V_{A}=-\frac{\sigma}{\varepsilon_{0}}\left(r_{B}-r_{A}\right)$$

(vi) Uniformly charged hollow conducting/ nonconducting/solid conducting sphere:

(a) $$\vec{E}=\frac{k Q}{|\vec{r}|^{2}} \hat{r}, r \geq R, V=\frac{K Q}{r}$$

(b) $$\vec{E}=0$$

for $$r<R, V=\frac{K Q}{R}$$

(vii) Uniformly charged solid nonconducting sphere (insulating material)

(a) $$\vec{E}=\frac{k Q}{|\vec{r}|^{2}} \hat{r} \text { for } r \geq R, V=\frac{K Q}{r}$$

(b)$$\vec{E}=\frac{K Q \vec{r}}{R^{3}}=\frac{\rho \vec{r}}{3 \varepsilon_{0}} \text { for } r \leq R,$$

$$ V=\frac{\rho}{6 \varepsilon_{0}}\left(3 R^{2}-r^{2}\right)$$

(viii) thin uniformly charged disc (surface charge density is $\sigma$ )

$$E_{\text {axis }}=\frac{\sigma}{2 \varepsilon_{0}}\left[1-\frac{x}{\sqrt{R^{2}+x^{2}}}\right]$$

$$ V_{\text {axis }}=\frac{\sigma}{2 \varepsilon_{0}}\left[\sqrt{R^{2}+x^{2}}-x\right]$$

• Work done by external agent in taking a charge $q$ from $A$ to $B$ is:

$$\left(W_{e x t}\right)_{A B}=q\left(V_B-V_A\right)$$

or $$\left(W_{e l}\right)_{A B}=q\left(V_A-V_B\right)$$

Electrostatic Potential Energy

• The electrostatic potential energy of a point charge: $$\mathrm{U}=\mathrm{qV}$$

• PE of the system: $$U = \frac{U_1+U_2+…}{2}=(U_{12}+U_{13}+…+U_{1n})+(U_{23}+U_{24}+…+U_{2n})+(U_{34}+U_{35}+…+U_{3n})…$$

• Energy Density: $$U=\frac{1}{2} \varepsilon \mathrm{E}^{2}$$

• Self Energy of a uniformly charged shell: $$U_{\text {self }}=\frac{K Q^{2}}{2 R}$$

• Self Energy of a uniformly charged solid non-conducting sphere:$$U_{\text {self }}=\frac{3 K Q^{2}}{5 R}$$

Electric Field Intensity Due to Dipole

(i) on the axis $\vec{E}=\frac{2 K \vec{P}}{r^{3}}$

(ii) on the equatorial position: $\vec{E}=-\frac{K \vec{P}}{r^{3}}$

(iii) Total electric field at general point $O(r, \theta)$ is $E_{r e s}=\frac{K P}{r^{3}} \sqrt{1+3 \cos ^{2} \theta}$

Potential Energy of an Electric Dipole in External Electric Field:

$$U=-\vec{p} \cdot \vec{E}$$

Electric Dipole in Uniform Electric Field :

$$\text { Torque } \vec{\tau}=\vec{\mathrm{p}} \times \vec{\mathrm{E}} ; \quad \vec{\mathrm{F}}=0$$

Electric Dipole in Non-uniform Electric Field:

$$\text { torque } \vec{\tau}=\vec{p} \times \vec{E} ; U=-\vec{p} \cdot \vec{E}, $$

$$\text { Net force }|F|=\left|p \frac{\partial E}{\partial r}\right|$$

Electric Potential Due to Dipole at General Point $(r, \theta)$ :

$$\mathrm{V}=\frac{\mathrm{P} \cos \theta}{4 \pi \varepsilon_{0} \mathrm{r}^{2}}=\frac{\vec{\mathrm{p}} \cdot \vec{\mathrm{r}}}{4 \pi \varepsilon_{0} \mathrm{r}^{3}}$$

Electric Flux:

• The electric flux over the whole area is given by: $$\phi_{E}=\int_{S} \vec{E} \cdot \vec{d S}=\int_{S} E_{n} d S$$

• Flux using Gauss’s law, Flux through a closed surface: $$\phi_{E}=\oint \vec{E} \cdot \vec{dS}=\frac{q_{in}}{\varepsilon_{0}}$$

• Electric field intensity near the conducting surface: $$\phi_{E}=\frac{\sigma}{\varepsilon_{0}} \hat{n}$$

Electric Pressure :

Electric pressure at the surface of a conductor is given by formula

$$P=\frac{\sigma^{2}}{2 \varepsilon_{0}}$$

where: $\sigma$ is the local surface charge density.

Work Done By An Electric Force:

$$W = q \vec{E} \cdot \vec{d}$$

Intensity:

$$I = \frac{1}{2} \epsilon_0 c |\vec{E}|^2$$