PiTheory

Electromagnetic Induction formulas

Class 12 physics formula sheet for NEET and JEE: the key equations of NCERT chapter 6, the special cases questions are built on, and diagrams where they help.

44 formulas8 sectionsClass 12 · Chapter 63 of 8 sections free

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

Most used formulasOther formulas and cases

1Magnetic flux

$$\phi=\vec B\cdot\vec A=BA\cos\theta,\qquad \phi=\int\vec B\cdot d\vec A$$
$$\begin{array}{l}\displaystyle \phi=\vec B\cdot\vec A=BA\cos\theta\\[6pt]\displaystyle \phi=\int\vec B\cdot d\vec A\end{array}$$

$\theta$ = angle between $\vec B$ and the normal. Weber (T m2); scalar, can be negative. Coil of $N$ turns: linkage $N\phi$.

$$\Delta\phi=NBA(\cos\theta_2-\cos\theta_1)$$

Rotating a coil: $0\to90^\circ$ gives $NBA$; $0\to180^\circ$ (flip) gives $2NBA$.

2Faraday and Lenz

$$e=-N\frac{d\phi}{dt},\qquad i=\frac eR,\qquad q=\frac{N\,\Delta\phi}{R}$$
$$\begin{array}{l}\displaystyle e=-N\frac{d\phi}{dt}\\[6pt]\displaystyle i=\frac eR\\[6pt]\displaystyle q=\frac{N\,\Delta\phi}{R}\end{array}$$

emf exists even in an open circuit; current only in a closed one. Induced charge does not depend on how fast the flux changes; emf and current do. $q=$ area under the $i$–$t$ graph.

$$e=-NA\cos\theta\frac{dB}{dt}\ \ \text{or}\ \ -NB\cos\theta\frac{dA}{dt}$$

Changing $B$ or area. Coil in a solenoid: $e=\mu_0nA\dfrac{di}{dt}$. Flux given as $\phi(t)$: differentiate. Heat in time $T$: $\displaystyle\int\frac{e^2}{R}dt$.

  • Lenz's law (energy conservation): the induced current opposes the change in flux. Magnet approaching a coil is repelled; leaving, attracted.
  • Magnet dropped through a closed ring: $a<g$; through a cut ring: $a=g$. Falling through a long metal pipe: reaches a terminal speed.
  • Current in a nearby straight wire increasing → induced current in a coplanar loop is opposite to it on the near side.

3Motional emf

$$e=Blv,\qquad e=(\vec v\times\vec B)\cdot\vec l$$

Zero if any two of $\vec B,\vec l,\vec v$ are parallel. Bent or curved rod: use the straight line joining its ends (effective length). Right-hand rule gives the + end.

$$i=\frac{Blv}{R},\quad F=\frac{B^2l^2v}{R},\quad P=\frac{B^2l^2v^2}{R}$$
$$\begin{array}{l}\displaystyle i=\frac{Blv}{R}\\[6pt]\displaystyle F=\frac{B^2l^2v}{R}\\[6pt]\displaystyle P=\frac{B^2l^2v^2}{R}\end{array}$$

Rod on rails: force needed to keep constant speed; mechanical power = Joule heat. Rod of resistance $r$: use $R+r$.

Rvl××××××e = Blv, F = B²l²v/R, P = B²l²v²/R
$$v_T=\frac{mgR}{B^2l^2},\qquad v_T=\frac{mgR\sin\theta}{B^2l^2}$$

Terminal speed of a rod/loop falling vertically, or sliding down smooth rails at $\theta$ ($B$ ⟂ rails). Rod pulled by a hanging mass $m$: same $v_T$; acceleration at $v_T/2$ is $g/2$ (massless rod).

$$v=v_0e^{-t/\tau},\qquad \tau=\frac{mR}{B^2l^2}$$

Rod given a push and left alone: slows exponentially; total distance $v_0\tau$.

$$e=\frac{\mu_0Iv}{2\pi}\ln\frac ba$$

Rod (from $a$ to $b$) moving parallel to a long current. Rectangular loop (side $b$ parallel to the wire, side $l$ away from it, near side at $x$) moving away: $e=\dfrac{\mu_0Ibv\,l}{2\pi x(x+l)}$.

Iabve = (μ₀Iv/2π) ln(b/a)
$$e=\tfrac12B\omega l^2$$

Rod rotating about one end (centre at higher/lower potential by the right-hand rule). About its middle: 0 between ends. Disc or spoked wheel: $\tfrac12B\omega R^2$ between axle and rim.

ωl××××e = ½ Bωl²(disc / wheel: same,any number of spokes)
  • Rod moving vertically with ends E–W cuts $B_H$: $e=B_Hlv$. Horizontal motion cuts $B_V$: $e=B_Vlv$. Ends N–S, falling: $e=0$.
  • Loop moving entirely inside a uniform field: net emf 0 (but opposite sides have emf).
  • Rails with a capacitor: $Q=CBlv$; with changing speed $i=CBl\dfrac{dv}{dt}$.

5 more sections and 26 formulas in the full chapter

  1. 4Induced electric field3 formulas · 1 diagram
  2. 5Self inductance7 formulas
  3. 6Mutual inductance and combinations7 formulas
  4. 7LR and CR circuits7 formulas · 1 diagram
  5. 8AC generator2 formulas

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