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Electric Charges and Fields formulas

Class 12 physics formula sheet for NEET and JEE: the key equations of NCERT chapter 1, the special cases questions are built on, and diagrams where they help.

56 formulas9 sectionsClass 12 · Chapter 13 of 9 sections free

By Sreeraj P, M.Sc Physics · 10+ years teaching NEET and JEE

Most used formulasOther formulas and cases

1Electric charge

$$q=\pm ne,\qquad e=1.6\times10^{-19}\ \text{C}$$

Quantised, conserved, invariant (same at any speed), additive (scalar). A body with $n_1$ protons and $n_2$ electrons carries $(n_1-n_2)e$.

  • Charging by friction, conduction (conduction precedes repulsion), induction (induction precedes attraction; the inducing body loses nothing). Repulsion is the sure test of charge.
  • Negatively charged body gains mass slightly; positive loses mass.
  • Charge induced on a dielectric: $q'=q\left(1-\dfrac1K\right)$.
  • 1 esu $=\dfrac{1}{3\times10^9}$ C; 1 emu $=10$ C. Dimension of charge [AT].

2Coulomb's law

$$F=\frac{1}{4\pi\varepsilon_0}\frac{q_1q_2}{r^2},\qquad \frac1{4\pi\varepsilon_0}=9\times10^9\ \text{N m}^2\text{C}^{-2}$$
$$\begin{array}{l}\displaystyle F=\frac{1}{4\pi\varepsilon_0}\frac{q_1q_2}{r^2}\\[6pt]\displaystyle \frac1{4\pi\varepsilon_0}=9\times10^9\ \text{N m}^2\text{C}^{-2}\end{array}$$

$\varepsilon_0=8.85\times10^{-12}$ C2 N−1 m−2. Central, conservative, obeys Newton's third law; valid for point charges at rest.

$$\vec F_{1}=\frac{q_1q_2}{4\pi\varepsilon_0}\,\frac{\vec r_1-\vec r_2}{|\vec r_1-\vec r_2|^3}$$

Vector form: force on $q_1$ at $\vec r_1$ due to $q_2$ at $\vec r_2$ (put signs of the charges in).

$$F_{\text{med}}=\frac{F_{\text{air}}}{K},\qquad K_1r_1^2=K_2r_2^2\ (\text{same }F)$$
$$\begin{array}{l}\displaystyle F_{\text{med}}=\frac{F_{\text{air}}}{K}\\[6pt]\displaystyle K_1r_1^2=K_2r_2^2\ (\text{same }F)\end{array}$$

In a medium of dielectric constant $K$. A slab of thickness $t$ in the gap acts like air of thickness $t\sqrt K$: $F=\dfrac{kq_1q_2}{(r-t+t\sqrt K)^2}$. Metal slab: $F=0$.

$$\text{Touch and separate: }q'=\frac{q_1+q_2}{2},\qquad \frac{F'}{F}=\frac{(q_1+q_2)^2}{4q_1q_2}$$
$$\begin{array}{l}\displaystyle \text{Touch and separate: }q'=\frac{q_1+q_2}{2}\\[6pt]\displaystyle \frac{F'}{F}=\frac{(q_1+q_2)^2}{4q_1q_2}\end{array}$$

Identical conducting spheres; for unlike charges use signs. Two like charged bodies can still attract when very close (induction).

  • Superposition: $\vec F=\vec F_1+\vec F_2+\cdots$. Equilateral triangle, three equal $q$: $F=\sqrt3\,\dfrac{kq^2}{a^2}$ on each.
  • $F_e/F_g\approx10^{36}$ for two protons/electrons.

3Equilibrium of charges

$$x=\frac{r}{1+\sqrt{q_2/q_1}}\ \ (\text{from }q_1)$$

Where a third charge stays at rest between two like charges $q_1$, $q_2$ a distance $r$ apart (same point as the null point).

$$q_3=-\frac{q_1q_2}{(\sqrt{q_1}+\sqrt{q_2})^2}$$

Charge needed at that point so that all three are in equilibrium. Along the line it is stable for axial moves only if $q_3$ is opposite; equilibrium of free charges is never stable (Earnshaw).

$$\tan\theta=\frac{F_e}{mg},\qquad q^2=16\pi\varepsilon_0\,l^2mg\tan\theta\,\sin^2\theta$$
$$\begin{array}{l}\displaystyle \tan\theta=\frac{F_e}{mg}\\[6pt]\displaystyle q^2=16\pi\varepsilon_0\,l^2mg\tan\theta\,\sin^2\theta\end{array}$$

Two equal balls on threads of length $l$, each at $\theta$ to the vertical. Immersed in liquid (density $\sigma$) with no change in angle: $K=\dfrac{\rho}{\rho-\sigma}$.

$$T=\sqrt{\frac{16\pi^3\varepsilon_0\,m r^3}{q_1q_2}}$$

Period of a charge $q_1$ (mass $m$) orbiting a fixed $q_2$ (electric force = centripetal force).

6 more sections and 42 formulas in the full chapter

  1. 4Electric field4 formulas
  2. 5Charged particle in a uniform field10 formulas · 1 diagram
  3. 6Continuous charge distributions8 formulas · 3 diagrams
  4. 7Electric flux and Gauss's law8 formulas · 2 case tables · 2 diagrams
  5. 8Conductors
  6. 9Electric dipole12 formulas · 1 diagram

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