Magnetic Effect Of Current And Magnetic Force On Charge
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:
PYQ-2023-Magnetic-Effects-Of-Current-Q4 , PYQ-2023-Magnetic-Effects-Of-Current-Q8, PYQ-2023-Magnetic-Effects-Of-Current-Q15, PYQ-2023-Magnetic-Effects-Of-Current-Q16, PYQ-2023-Magnetic-Effects-Of-Current-Q19, PYQ-2023-Magnetic-Effects-Of-Current-Q20, PYQ-2023-Magnetic-Effects-Of-Current-Q23
$$B=\frac{\mu_{0}}{4 \pi} \frac{I}{r}\left(\sin \theta_{1}+\sin \theta_{2}\right)$$
Magnetic Field Due To Infinite Straight Wire:
PYQ-2023-Magnetic-Effects-Of-Current-Q2, PYQ-2023-Magnetic-Effects-Of-Current-Q5, PYQ-2023-Magnetic-Effects-Of-Current-Q9, PYQ-2023-Magnetic-Effects-Of-Current-Q10, PYQ-2023-Magnetic-Effects-Of-Current-Q11, PYQ-2023-Magnetic-Effects-Of-Current-Q12, PYQ-2023-Magnetic-Effects-Of-Current-Q13, PYQ-2023-Magnetic-Effects-Of-Current-Q19, PYQ-2023-Magnetic-Effects-Of-Current-Q20, PYQ-2023-Magnetic-Effects-Of-Current-Q21, PYQ-2023-Magnetic-Effects-Of-Current-Q22
$$B=\frac{\mu_{0}}{2 \pi} \frac{I}{r}$$
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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:
PYQ-2023-Magnetic-Effects-Of-Current-Q3, PYQ-2023-Magnetic-Effects-Of-Current-Q6
$$ \mathrm{B}=\frac{\mu_{0} \mathrm{nl}}{2}\left(\cos \theta_{1}-\cos \theta_{2}\right) $$
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:
PYQ-2023-Magnetic-Effects-Of-Current-Q14
$$\mathrm{M}=\mathrm{N} \cdot \mathrm{I} \cdot \mathrm{A}$$
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:
PYQ-2023-Magnetic-Effects-Of-Current-Q1
$$\frac{F}{L} = \frac{\mu_0 I_1 I_2}{2\pi d}$$
Moving Coil Galvanometer:
PYQ-2023-Magnetic-Effects-Of-Current-Q7, PYQ-2023-Current-Electricity-Q18
$$\theta = \frac{NBIA}{k}$$
Potential Energy Of A Magnetic Dipole:
PYQ-2023-Magnetic-Effects-Of-Current-Q18
$$U = - \vec{\mu} \cdot \vec{B}$$