Magnetic Effect of Current Formulas

1. Biot-savart’s law

The magnetic field at a certain point due to an element δl of a current-carrying conductor is

δB = $\frac{{\mu }_0}{4\pi }\frac{i\delta \ell \sin \theta }{r^2}$
or $\overrightarrow{\mathrm{d}\mathrm{B}}=\frac{{\mu }_0}{4\pi }\frac{\mathrm{i}\delta \overrightarrow{\ell }\times \hat{\mathrm{r}}}{{\mathrm{r}}^2}$
= $\frac{{\mu }_0}{4\pi }\frac{i\delta \overrightarrow{\ell }\times \overrightarrow{r}}{r^3}$
δ$\overrightarrow{\mathrm{B}}$ is in a direction normal to the plane of δ$\overrightarrow{\ell }\text{ and }\overrightarrow{r}$

2. Magnetic field due to a current caryying circular coil

(a) At the centre
B₀ = $\frac{{\mu }_0\mathrm{ni}}{2\mathrm{R}}$ along the axis of coil.

(b) At a point on the axis of a coil
B = $\frac{{\mu }_{0}ni{R}^2}{2{(R^2+x^2)}^{3/2}}$

(c) If x > > R, then
B = $\frac{{\mu }_0{\mathrm{niR}}^2}{2{\mathrm{x}}^3}$

(d) At the point of inflexion, $\frac{\mathrm{dB}}{\mathrm{dx}}$ = constant or $\frac{d^{2}B}{d{x}^2}$ inflection are found in the field of a coil at x = ± R/2 and the distance between them is equal to the radius of the coil.

3. Magnetic field due to a current carrying straight wire of finite length

B = $\frac{{\mu }_0\mathrm{i}}{4\pi \mathrm{R}}$ (sin β₁ + sin β₂)
or B = $\frac{{\mu }_0\mathrm{i}}{4\pi \mathrm{R}}$ (cos α₁ + cos α₂)

4. Magnetising field ($\overrightarrow{\mathrm{H}}$)

$\overrightarrow{\mathrm{H}}=\overrightarrow{\mathrm{B}}/{\mu }_0$

5. Ampere’s law

(a) The line integral of magnetic field along the closed path = p0 multiple of net current passing through that closed path
$\oint \overrightarrow{\mathrm{B}}\cdot \mathrm{d}\overrightarrow{\ell }={\mu }_0\mathrm{\Sigma }\mathrm{I}$

(b) Magnetomotive force
F_m = $\oint \overrightarrow{\mathrm{H}}\cdot \mathrm{d}\overrightarrow{\ell }=\frac{1}{\mu }\oint \overrightarrow{\mathrm{B}}\cdot \mathrm{d}\overrightarrow{\ell }$

6. Magnetic field due to a current carrying straight wire of infinite length

B = $\frac{{\mu }_{0}i}{2\pi r}=\frac{{\mu }_0}{4\pi }\frac{2i}{r}$

7. Magnetic field due to a current carrying long and straight solid cylinder

• At a point out side the cylinder B_out = $\frac{{\mu }_{0}i}{2\pi r}$
• At the surface B_surface = $\frac{{\mu }_{0}i}{2\pi R}$
• At a point inside the cylinder B_in = $\frac{{\mu }_0\mathrm{ir}}{2\pi {\mathrm{R}}^2}$ ; R → Radius of cylinder.

8. Magnetic field due to a current carrying long and straight hollow cylinder

(a) At a point out side the cylinder
B_out = $\frac{{\mu }_{0}i}{2\pi r}$

9. Magnetic behaviour of current carrying coil and its magnetic moment

• A small current carrying coil behaves like a small magnet.
• Magnetic moment of a current carrying coil

M = current × effective area.
For a coil of N turns
M = NiA = NiπR²

10. Current and magnetic field due to circular motion of charge

(a) Current i = ef = $\frac{\mathrm{e}}{\mathrm{T}}$
f → revolution/second, T → Time period
i = $\frac{\mathrm{e}\omega }{2\pi }=\frac{\mathrm{ev}}{2\pi \mathrm{R}}$

(b) Magnetic field B₀ = $\frac{{\mu }_{0}nI}{2R}=\frac{{\mu }_{0}nef}{2R}=\frac{{\mu }_{0}ne}{2RT}$
B₀ = $\frac{{\mu }_0\text{ ne }\omega }{4\pi R}=\frac{{\mu }_0\text{ nev }}{4\pi R^2}$ (e → charge of electron)

(c) Magnetic moment
M = iA = efπR² = $\frac{\mathrm{e}\pi {\mathrm{R}}^2}{\mathrm{T}}$
M = $\frac{\mathrm{e}\omega {\mathrm{R}}^2}{2}=\frac{\mathrm{evR}}{2}=\frac{\mathrm{eL}}{2\mathrm{m}}$
L → angular momentum, m → mass of electron

11. Force on a current carrying condcutor due to magnetic field

$\overrightarrow{\mathrm{F}}=\mathrm{i}(\overrightarrow{\ell }\times \overrightarrow{\mathrm{B}})$
$|\overrightarrow{\mathrm{F}}|$ = i l B sin θ
Two parallel conductors carrying currents in the same direction attract each other but with currents in opposite direction repel each other.

12. Magnetic force between two parallel current-carrying conductors

Attractive or repulsive force on unit length of conductors
$\frac{F}{\ell }=\frac{{\mu }_0{i}_1{i}_2}{2\pi d}$
d → distance between parallel conductors.

13. Motion of charged particle in a magnetic field

(a) Force on the particle
$\overrightarrow{\mathrm{F}}=\mathrm{q}(\overrightarrow{\mathrm{v}}\times \overrightarrow{\mathrm{B}})$
$|\overrightarrow{\mathrm{F}}|$ = qvB sin θ

(b) when θ = 90°, the motion of particle will be along a circular path. Radius of circular path
R = $\frac{mv}{qB}=\frac{\sqrt{2mE}}{qB}=\frac{\sqrt{2mqV}}{qB}$
Period of revolution of the particle
T = $\frac{2\pi \mathrm{m}}{\mathrm{qB}}$
Frequency of revolution
f = $\frac{1}{\mathrm{T}}=\frac{\mathrm{qB}}{2\pi \mathrm{m}}$
Kinetic energy of the particle
E = $\frac{{\mathrm{R}}^2{\mathrm{q}}^2{\mathrm{B}}^2}{2\mathrm{m}}$

14. Interaction between two moving charges

(a) Magnetic field due to charge moving with velocity $\overrightarrow{\mathrm{v}}$
$\overrightarrow{\mathrm{B}}=\frac{{\mu }_0}{4\pi }\frac{\mathrm{q}(\overrightarrow{\mathrm{v}}\times \overrightarrow{\mathrm{r}})}{{\mathrm{r}}^3}$
Hence B = $\frac{{\mu }_0}{4\pi }\frac{qv\sin \theta }{r^2}$

(b) The electric and magnetic forces both act between moving charges.

(c) Electric force F_e = $\frac{1}{4\pi {ϵ}_0}\frac{q_1{q}_2}{r^2}$

(d) Magnetic force F_m = $\frac{{\mu }_0}{4\pi }\frac{{\mathrm{q}}_1{\mathrm{q}}_2{\mathrm{v}}_1{\mathrm{v}}_2}{{\mathrm{r}}^2}$
If v₁ = v₂ = v
then F_m = $\frac{{\mu }_0}{4\pi }\frac{q_1{q}_2}{r^2}{v}^2$

(e) $\frac{F_m}{F_e}=\frac{v^2}{c^2}={\left(\frac{v}{c}\right)}^2$
Stationary Charges:

Moving Charges:

15. Force and torque on a current-carrying coil placed in a uniform magnetic field

(a) resultant force F_net = 0.

(b) A torque acts on the coil
τ = iNAB sin θ = MB sin θ
M → magnetic dipole moment.
In vector form
τ = $\overrightarrow{\mathrm{M}}\times \overrightarrow{\mathrm{B}}$

(c) The work done in turning a loop from angle θ₁ to θ₂.
W = MB (cos θ₁ – cos θ₂)

(d) Time period of oscillation of a magnetic dipole in uniform M.F.
T = 2π$\sqrt{\frac{\mathrm{I}}{\mathrm{MB}}}$; I → moment of inertia

Get instant help regarding formulas of various concepts from Physics all at one place on Onlinecalculator.guru a trusted and reliable portal.