Magnetic Field Due To A Moving Point Charge:

$$ \vec{B}=\frac{\mu_{0}}{4 \pi} \cdot \frac{q(\vec{v} \times \vec{r})}{r^{3}} $$

Biot-Savart’s Law:

$$ \overrightarrow{\mathrm{dB}}=\frac{\mu_{0} l}{4 \pi} \cdot\left(\frac{\overrightarrow{\mathrm{d} \ell} \times \overrightarrow{\mathrm{r}}}{\mathrm{r}^{3}}\right) $$

Magnetic Field Due To A Straight Wire:

,

$$B=\frac{\mu_{0}}{4 \pi} \frac{I}{r}\left(\sin \theta_{1}+\sin \theta_{2}\right)$$

Magnetic Field Due To Infinite Straight Wire:

$$B=\frac{\mu_{0}}{2 \pi} \frac{I}{r}$$

• Magnetic Field Due To Circular Loop

  (i) At centre

  

  $$ B=\frac{\mu_{0} N I}{2 r} $$

  (ii) At Axis

  $$ B=\frac{\mu_{0}}{2}\left(\frac{N I R^{2}}{\left(R^{2}+x^{2}\right)^{3 / 2}}\right) $$

Magnetic Field On The Axis Of The Solenoid:

$$ \mathrm{B}=\frac{\mu_{0} \mathrm{nl}}{2}\left(\cos \theta_{1}-\cos \theta_{2}\right) $$

Ampere’s Law:

$$\oint \vec{B} \cdot d \vec{\ell}=\mu_0 l$$

Magnetic Field Due To Long Cylindrical Shell:

$$ \begin{aligned} & B=0, r<R \\ & =\frac{\mu_{0}}{2 \pi} \frac{I}{r}, r \geq R \end{aligned} $$

Magnetic Force Acting On A Moving Point Charge:

a. $$ \overrightarrow{\mathrm{F}}=\mathrm{q}(\vec{v} \times \overrightarrow{\mathrm{B}})$$

$$ \begin{aligned} (i) & &\vec{v} \perp \vec{B} \\ & & r=\frac{m v}{q B}\\ & & \mathrm{T}=\frac{2 \pi \mathrm{m}}{\mathrm{qB}} \end{aligned} $$

(ii) $$\quad$$

$$ r=\frac{m v \sin \theta}{q B} $$

$$ \mathrm{T}=\frac{2 \pi \mathrm{m}}{\mathrm{qB}} $$

Pitch $$=\frac{2 \pi \mathrm{m} v \cos \theta}{\mathrm{qB}}$$

b. $$ \vec{F}=q ((\vec{v} \times \vec{B})+\vec{E})$$

Magnetic Force Acting On A Current Carrying Wire:

$$ \overrightarrow{\mathrm{F}}=I(\vec{\ell} \times \vec{B}) $$

Magnetic Moment Of A Current Carrying Loop:

$$\mathrm{M}=\mathrm{N} \cdot \mathrm{I} \cdot \mathrm{A}$$

Torque Acting On A Loop:

$$\vec{\tau}=\vec{M} \times \vec{B}$$

Magnetic field due to a single pole:

$$B=\frac{\mu_{0}}{4 \pi} \cdot \frac{m}{r^{2}}$$

Magnetic field on the axis of magnet:

$$B=\frac{\mu_{0}}{4 \pi} \cdot \frac{2 M}{r^{3}}$$

Magnetic Field On The Equatorial Axis Of The Magnet:

$$B=\frac{\mu_{0}}{4 \pi} \cdot \frac{M}{r^{3}}$$

Magnetic Field At Point P Due To Magnet:

$$B=\frac{\mu_{0}}{4 \pi} \frac{M}{r^{3}} \sqrt{1+3 \cos ^{2} \theta}$$

The Force Per Unit Length Between Two Parallel Wires Carrying Currents:

$$\frac{F}{L} = \frac{\mu_0 I_1 I_2}{2\pi d}$$

Moving Coil Galvanometer:

$$\theta = \frac{NBIA}{k}$$

Potential Energy Of A Magnetic Dipole:

$$U = - \vec{\mu} \cdot \vec{B}$$