Magnetism Formulas

1. Magnetic moment

(a) The magnetic moment produced due to motion of electron:
M = iA = – \(\frac{\mathrm{ev}}{2 \pi \mathrm{r}}\) (╧Çr²) = – \(\frac{\mathrm{evr}}{2}=-\frac{\mathrm{e} \omega \mathrm{r}^{2}}{2}\)
= \(\frac{\mathrm{e}}{2 \mathrm{m}}\)L = -n\(\left(\frac{\mathrm{eh}}{2 \pi \mathrm{m}}\right)\),
Here ┬╡_B = \(\frac{\mathrm{eh}}{2 \pi \mathrm{m}}\) Bohr’s magneton
where L is the angular momentum of the electron,

(b) For a bar magnet
M = m × 2l
where m is the pole strength of the magnet.

2. Magnetic field due to a bar magnet

On the point of axis
B = \(\frac{\mu_{0} 2 \mathrm{Mr}}{4 \pi\left(\mathrm{r}^{2}-\ell^{2}\right)^{2}}\)
B Γëê \(\frac{\mu_{0}}{4 \pi} \frac{2 M}{r^{3}}\) (when r >> l)

On the point of equatorial
B = \(\frac{\mu_{0}}{4 \pi} \frac{M}{\left(r^{2}+\ell^{2}\right)^{3 / 2}}\)
B = \(\frac{\mu_{0}}{4 \pi} \frac{M}{r^{3}}\) when r >> l
At a general point

3. Intensity of magnetisation (\(\overrightarrow{\mathrm{I}}\))

The magnetic moment induced per unit volume of the substance is intensity of magnetisation.
(a) I = \(\frac{M}{V}\) A/m
(b) I = \(\frac{m}{A}\)
A → cross sectional area; m → pole strength

4. Magnetic susceptibility (χ)

I ∝ H
I = χH
or χ = \(\frac{1}{\mathrm{H}}\) unitless quantity.

5. Magnetic permeability (┬╡)

i.e., ┬╡ = \(\frac{B}{H}\)
For vacuum ┬╡₀ = \(\frac{\mathrm{B}_{0}}{\mathrm{H}}\) = 4╧Ç ├ù 10^-7 H/m
Relative magnetic permeability
┬╡_r = \(\frac{\mu}{\mu_{0}}\)

6. Relations amongst B, H, I, χ AND µ

• B = ┬╡H = B₀ + ┬╡₀I = ┬╡₀(H + I)
• B = ┬╡₀H(l + ╧ç)
• ┬╡ = ┬╡₀(1 + ╧ç)
• ┬╡_r = (1 + ╧ç)

7. Magnetic hysterisis loop, B-H or I-H curve

• B always lags behind H. This property is called magnetic hysterics.
• The curve drawn between B and H is called B-H curve. It is a closed curve for one complete cycle of magnetisation and demagnetisation.
• Hysterisis loss per cycle per unit volume = area bound by the hysterics loop.

8. Types of magnetic materials

• Diamagnetic material (┬╡ < 1)
• Paramagnetic material (┬╡ > l)
• Ferromagnetic material (┬╡ >> 1)

9. CurieΓÇÖs law and curie temperature

(a) For a paramagnetic material
χ ∝ \(\frac{1}{T}\) or χ = \(\frac{C}{T}\)

(b) For a ferromagnetic material
χ = \(\frac{C}{\left(T-T_{c}\right)}\)
where T_C is curie temperature.

(c) The temperature at or above which the ferromagnetic material behaves like a paramagnetic material is called Curie temperature.

10. Terrestrial magnetism

Deals with the magnetic field of earth. It is described in terms of three quantities, which are called magnetic elements of earth. These are:
(i) Magnetic declination (╬╕) at a place is the angle ╬╕ between magnetic axis and the geographic axis. It is also equal to the angle between magnetic meridian and geographic merdian at the place

(ii) Magnetic Inclination or dip at a place is defined as the angle ╬┤, which the direction of total intensity of earthΓÇÖs magnetic field, (B) makes with a horizontal line in magnetic meridian.

(iii) Horizontal component, H is the component of total intensity of earth’s magnetic field (B) in the horizontal direction in magnetic meridian,
H = B cos ╬┤
V = B sin ╬┤
∴ \(\sqrt{H^{2}+V^{2}}\) = B
and \(\frac{V}{H}\) = tan ╬┤

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