PiTheory

Magnetism and Matter formulas

Class 12 physics formula sheet for NEET and JEE: the key equations of NCERT chapter 5, the special cases questions are built on, and diagrams where they help.

37 formulas8 sectionsClass 12 · Chapter 53 of 8 sections free

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

Most used formulasOther formulas and cases

1Bar magnet and magnetic moment

$$M=m\times2l,\qquad \vec M:\ S\to N$$

Pole strength $m$ (A m); moment (A m2). Magnetic length $2l\approx\dfrac56\times$ geometric length. Poles always come in pairs (no monopoles); solenoid and toroid behave as magnets.

$$\text{cut into }n\text{ parts}\ \perp\text{ length: }m,\ \frac Mn;\qquad \parallel\text{ length: }\frac mn,\ \frac Mn$$
$$\begin{array}{l}\displaystyle \text{cut into }n\text{ parts}\ \perp\text{ length: }m\\[6pt]\displaystyle \frac Mn;\\[6pt]\displaystyle \parallel\text{ length: }\frac mn,\ \frac Mn\end{array}$$

$x$ parts along and $y$ across: each part $\dfrac{M}{xy}$ with pole strength $m/x$.

$$\text{bent into an arc }\theta:\ M'=\frac{2M\sin(\theta/2)}{\theta}$$

Pole strength unchanged, length becomes the chord. Semicircle $\dfrac{2M}{\pi}$; quadrant $\dfrac{2\sqrt2M}{\pi}$; circle 0. Bent at the middle at $90^\circ$: $\dfrac{M}{\sqrt2}$; into a "U" of three equal parts: $\dfrac M3$.

SNnew length 2R sin(θ/2)θ
$$M_{\text{net}}=\sqrt{M_1^2+M_2^2+2M_1M_2\cos\theta}$$

$\theta$ = angle between the two moments (like poles touching → angle between axes; unlike poles → $180^\circ-$ that). Closed polygon of identical magnets, unlike poles at corners: 0; one reversed: $2M$.

2Field of a magnet

$$B_{\text{axial}}=\frac{\mu_0}{4\pi}\frac{2M}{d^3},\qquad B_{\text{eq}}=\frac{\mu_0}{4\pi}\frac{M}{d^3}$$
$$\begin{array}{l}\displaystyle B_{\text{axial}}=\frac{\mu_0}{4\pi}\frac{2M}{d^3}\\[6pt]\displaystyle B_{\text{eq}}=\frac{\mu_0}{4\pi}\frac{M}{d^3}\end{array}$$

Short magnet. Long magnet: $\dfrac{\mu_0}{4\pi}\dfrac{2Md}{(d^2-l^2)^2}$ and $\dfrac{\mu_0}{4\pi}\dfrac{M}{(d^2+l^2)^{3/2}}$. General point: $\dfrac{\mu_0}{4\pi}\dfrac{M}{d^3}\sqrt{1+3\cos^2\theta}$.

$$B=\frac{\mu_0}{4\pi}\frac{m}{r^2},\qquad F=mB$$

Isolated pole; force on a pole in a field.

$$F=\frac{\mu_0}{4\pi}\frac{6M_1M_2}{r^4}\ (\text{coaxial}),\qquad \frac{\mu_0}{4\pi}\frac{3M_1M_2}{r^4}\ (\text{side by side})$$
$$\begin{array}{l}\displaystyle F=\frac{\mu_0}{4\pi}\frac{6M_1M_2}{r^4}\ (\text{coaxial})\\[6pt]\displaystyle \frac{\mu_0}{4\pi}\frac{3M_1M_2}{r^4}\ (\text{side by side})\end{array}$$

Force between two short magnets.

3Magnet in a uniform field

$$\tau=MB\sin\theta,\quad U=-MB\cos\theta,\quad W=MB(\cos\theta_1-\cos\theta_2)$$
$$\begin{array}{l}\displaystyle \tau=MB\sin\theta\\[6pt]\displaystyle U=-MB\cos\theta\\[6pt]\displaystyle W=MB(\cos\theta_1-\cos\theta_2)\end{array}$$

Only a couple (no net force) in a uniform field; non-uniform field → force too. Torque needed to hold at $\theta$ after work $W$ from equilibrium: $\tau=W\cot(\theta/2)$ at $60^\circ$ gives $\sqrt3W$.

$$T=2\pi\sqrt{\frac{I}{MB_H}},\qquad I=\frac{m_0(l^2+b^2)}{12}$$
$$\begin{array}{l}\displaystyle T=2\pi\sqrt{\frac{I}{MB_H}}\\[6pt]\displaystyle I=\frac{m_0(l^2+b^2)}{12}\end{array}$$

Vibration magnetometer. $T\propto1/\sqrt{MB}$. $\dfrac{\Delta T}{T}=\tfrac12\dfrac{\Delta I}{I}=-\tfrac12\dfrac{\Delta M}{M}$.

$$\frac{M_1}{M_2}=\frac{T_2^2+T_1^2}{T_2^2-T_1^2}$$

Two magnets together: like poles ($T_1$), one reversed ($T_2$). Cut ⟂ into $n$: $T'=T/n$; cut along length: $T$ unchanged. Identical unmagnetised bar added: $T\sqrt2$.

$$\frac{T}{T_0}=\sqrt{\frac{B_H}{B_H\pm B}}$$

Extra field $B$ (another magnet or a wire $\mu_0I/2\pi r$) aiding (+) or opposing (−) $B_H$; perpendicular: $\sqrt{B_H^2+B^2}$.

$$M\!B\sin\theta=C(\alpha-\theta)$$

Magnet on a suspension wire twisted by $\alpha$ turns through $\theta$ ($C$ = torque per unit twist).

5 more sections and 17 formulas in the full chapter

  1. 4Neutral points4 formulas
  2. 5Gauss's law and magnetic quantities7 formulas
  3. 6Dia, para and ferromagnetism1 case table
  4. 7Hysteresis1 diagram
  5. 8Earth's magnetism6 formulas · 1 diagram

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