1. Hollow Conducting Sphere Electric Field:

(i) Inside the sphere (r < R) $$E=0$$

(ii) Outside the sphere (r > R) $$E=\dfrac{1}{4\pi \epsilon_0}\dfrac{Q}{r^2}$$

2. Solid Conducting Sphere Electric Field:

(i) Inside the sphere (r < R) $$E=\dfrac{1}{4\pi \epsilon_0}\dfrac{Qr}{R^3}$$

(ii) Outside the sphere (r > R) $$E=\dfrac{1}{4\pi \epsilon_0}\dfrac{Q}{r^2}$$

3. Concentric Conducting Spheres Electric Field:

(i) Inside the inner sphere (r < R1) $$E=0$$

(ii) Between the spheres (R1 < r < R2) $$E=\dfrac{1}{4\pi \epsilon_0}\dfrac{Q_1}{r^2}$$

(iii) Outside the outer sphere (r > R2) $$E=\dfrac{1}{4\pi \epsilon_0}\dfrac{Q_1+Q_2}{r^2}$$

4. Cylindrical Capacitor Electric Field:

(i) Inside the inner cylinder (r < R1) $$E=0$$

(ii) Between the cylinders (R1 < r < R2) $$E=\dfrac{1}{2\pi \epsilon_0}\dfrac{Q_1}{r}$$

5. Parallel-Plate Capacitor Electric Field: $$E=\dfrac{\sigma}{\epsilon_0}=\dfrac{Q}{Ad\epsilon_0}$$

---

CBSE BOARD NUMERICALS#

---

1. Conducting Sphere (CBSE Board) Electric Field:

$$E=\dfrac{1}{4\pi \epsilon_0}\dfrac{Q}{r^2}\text{ for }r>R$$

2. Two Point Charges Electric Field: $$E=\dfrac{1}{4\pi \epsilon_0}\left[\dfrac{Q_1}{(r-s)^2}+\dfrac{Q_2}{s^2}\right]$$

3. Dipole Electric Field: $$E=\dfrac{1}{4\pi \epsilon_0}\dfrac{2Q d}{r^3}$$

4. Uniform Electric Field Electric Field: $$\vec{E}=\vec{E}Q+\vec{E}{uniform}$$ $$=\dfrac{1}{4\pi \epsilon_0}\dfrac{Q}{r^2}\hat{r}+E\hat{i}$$

5. Current-Carrying Wire Magnetic Force: $$F=I L B \sin \theta \hat{n}$$