Shortcut Methods
JEE NUMERICALS
- Hollow Conducting Sphere Electric Field:
(i) Inside the sphere (r < R) $$E=0$$
(ii) Outside the sphere (r > R) $$E=\dfrac{1}{4\pi \epsilon_0}\dfrac{Q}{r^2}$$
- 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) $$E=\dfrac{1}{4\pi \epsilon_0}\dfrac{Q}{r^2}$$
- 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) $$E=\dfrac{1}{4\pi \epsilon_0}\dfrac{Q_1+Q_2}{r^2}$$
- Cylindrical Capacitor Electric Field:
(i) Inside the inner cylinder (r < R1) $$E=0$$
(ii) Between the cylinders (R1 < r < R2) $$E=\dfrac{1}{2\pi \epsilon_0}\dfrac{Q_1}{r}$$
- Parallel-Plate Capacitor Electric Field: $$E=\dfrac{\sigma}{\epsilon_0}=\dfrac{Q}{Ad\epsilon_0}$$
CBSE BOARD NUMERICALS
- Conducting Sphere (CBSE Board) Electric Field:
$$E=\dfrac{1}{4\pi \epsilon_0}\dfrac{Q}{r^2}\text{ for }r>R$$
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Two Point Charges Electric Field: $$E=\dfrac{1}{4\pi \epsilon_0}\left[\dfrac{Q_1}{(r-s)^2}+\dfrac{Q_2}{s^2}\right]$$
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Dipole Electric Field: $$E=\dfrac{1}{4\pi \epsilon_0}\dfrac{2Q d}{r^3}$$
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Uniform Electric Field Electric Field: $$\vec{E}=\vec{E}Q+\vec{E}{uniform}$$ $$=\dfrac{1}{4\pi \epsilon_0}\dfrac{Q}{r^2}\hat{r}+E\hat{i}$$
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Current-Carrying Wire Magnetic Force: $$F=I L B \sin \theta \hat{n}$$
