## Magnetism and matter

# The Bar Magnet, Magnetism and Gauss's Law

- A piece of matter, which when suspended freely rests itself in a particular direction (north-south) and which possesses a net magnetic moment and which attracts pieces of ferromagnetic materials like iron, cobalt, Nickel etc towards it, is called a MAGNET.
- A piece of magnet is dipped in iron fillings, they cling to it, especially at certain points. These points are called the poles of the magnet.
- The straight line joining the two poles of a bar magnet is called axial line.
- The line passing through the midpoint normal to the axial line is called equatorial line.
- The force between two isolated magnetic poles is expressed by \tt F=\frac{\mu_{0}}{4\pi}\ \frac{m_{1}m_{2}}{d^{2}}
- \tt F_{Medium}=\frac{\mu}{4\pi}\ \frac{m_{1}m_{2}}{d^{2}}. When the poles are in air or vacuum.

μ_{0}= permeability of free space

m_{1}, m_{2}= pole strengths of the pole

SI unit of pole strength = A-m - Relative permeability of the medium

\tt \mu_{r}=\frac{Permeability\ of\ the\ medium}{Permeability\ of\ air\ or\ vacuum}\ \frac{\mu}{\mu_{0}}

μr has no units and no dimensions - Two magnetic poles of same strength (m) are placed at two vertices of an equilateral triangle of side ‘d’ the force on a similar magnetic pole on third corner is given by \tt F^{1}=\sqrt{3}\ \frac{\mu_{0}}{4\pi}\ \frac{m^{2}}{a^{2}}
- Two unlike magnetic poles of same strength (m) are placed at two vertical of an equilateral triangle of side ‘d’. If a north pole of same strength is placed at the third vertex, it experiences a force of magnitude which is given by \tt F^{1}= \frac{\mu_{0}}{4\pi}\ \frac{m^{2}}{a^{2}}
- Four like poles each of pole strength ‘m’ are kept at the four corners of a square of side ‘d’ The net magnetic force on the pole at any one corner is given by \tt F^{1}= \frac{\mu_{0}}{4\pi}\ \frac{m^{2}}{a^{2}}\left[\sqrt{2}+\frac{1}{2}\right]
- Force between two co-axial magnetic dipoles \tt F= \frac{\mu_{0}}{4\pi}\ \frac{6\ M_{1}M_{2}}{d^{4}}
- The force between two parallel magnets \tt F= \frac{\mu_{0}}{4\pi}\ \frac{3\ M_{1}M_{2}}{d^{4}}
- The distance between the two poles of a magnet is called magnetic length of a magnet. It is generally represented as ‘2l’.
- It is directed along the south pole to north pole and it is a vector.
- Magnetic length = \tt \frac{5}{6} × geometric length
- Magnetic Moment (M) is measured as the product of magnetic length and pole strength. M = 2l × m.
- When a bar magnet is cut into ‘n’ equal parts parallel to its axis

Pole strength of each part = M^{1}= M/n

Length of each part = (2l)^{1}= 2l

Magnetic moment of each part M^{1}= M/n - When a bar magnet is cut into ‘n’ equal parts normal to its axis

Pole strength of each part = M^{1}= M

Length of each part = 2 l/n

Magnetic moment of each part M^{1}= M/n - When a bar magnet is cut into ‘xy’ equal parts ‘x’ parts parallel to its axis and ‘y’ parts normal to the axis.
- Pole strength of each part M
^{1}= m/x - Length of each part (2l)
^{1}= 2l/y - Magnetic moment of each part M
^{2}=M/xy - When two magnets of magnetic moments M
_{1}and M_{2}are kept at an angle ‘θ’ with like poles touching each other, then the resultant magnetic moment

\tt M^{1}=\sqrt{M_{1}^{2}+M_{2}^{2}+2M_{1}M_{2}\cos \theta} - When two magnets of magnetic moments M
_{1}and M_{2}are kept at an angle ‘θ’ with unlike poles touching each other, then the resultant magnetic moment.

\tt M^{1}=\sqrt{M_{1}^{2}+M_{2}^{2}-2M_{1}M_{2}\cos \theta} - On bending a magnet its pole strength remains unchanged where as its magnetic moment changes.
- When a thin bar magnet of magnetic moment ‘M’ and length ‘2l’ is bent oil its midpoint with and angle ‘θ’ between the two parts its effective length becomes 2l Sin θ/2
- Its new magnetic moment \tt M^{1}=M\ \sin\frac{\theta}{2}
- When a magnetised wire of length ‘2l’ and magnetic moment ‘M’ is bent in to an arc of a circle, that makes an angle ‘θ’ at the centre of the circle. Its magnetic moment decreases and becomes

\tt M^{1}=\left[\frac{2\ M\ \sin\left(\theta/2\right)}{\theta}\right] where θ is in radian - Like in electric field, the intensity of magnetic induction at a point in a magnetic field is defined as force experienced by a unit north pole placed at that point in magnitude as well as direction.

\tt \overrightarrow{B}=\frac{\overrightarrow{F}}{m},B=\frac{\mu_{0}}{4\pi}\cdot\frac{m}{d^{2}} - S.I unit \tt \overrightarrow{B} is Tesla or weber /m
^{2} - Dimensional Formula of B is [MT
^{−2}I^{−2}] - The magnetic induction at the centre of the line joining the two poles of a horse shoe magnet of pole strength ‘m’ and separated by a distance ‘d’ is

\tt B=8\ \frac{\mu_{0}}{4\pi}\ \frac{m}{d^{2}} directed from N to S pole - A magnetic line of force is the path followed by a free unit north pole in a magnetic field. Magnetic field can be represented by magnetic lines of force. They are imaginary.
- Two magnetic lines of force never intersect; if they intersect it means that field has two directions at the point of intersection which is impossible.
- The magnetic field, in which the magnetic induction field strength is same both in magnitude and direction at all points, is known as UNIFORM MAGNETIC FIELD.
- The magnetic field, in which the magnetic induction or field strength differs either in magnitude or in direction or both is known as NON UNIFORM MAGNETIC FIELD.
- Magnetic flux (Φ) linked through a cross section is defined as dot product of \tt \overrightarrow{B}\ and\ \overrightarrow{A}\ \ \phi=\overrightarrow{B}\cdot\overrightarrow{A}\ =BA\ \cos \theta

Where θ = angle made by magnetic field \tt \overrightarrow{B} with the normal to the area \tt \left(\hat{n}\right)\ \overrightarrow{A}=A\hat{n} A = area of the coil - B at a point ‘P’ on the axial line of a bar magnet at a distance ‘d’ from the centre of the magnet is \tt B_{a}=\frac{\mu_{0}}{4\pi}\ \frac{2Md}{\left(d^{2}-l^{2}\right)^2}
- B at a point ‘P’ on the equatorial line of a bar magnet at a distance ‘d’ from the centre of the magnet is \tt B_{e}=\frac{\mu_{0}}{4\pi}\ \frac{M}{\left(d^{2}+l^{2}\right)^3/2}
- In cast of short magnet \tt B_{e}=\frac{\mu_{0}}{4\pi}\frac{M}{d^3}\ \left(\therefore l<<d\right)
- Magnetic Induction at a points ‘P’ and whose position vector from the mid point of the magnet makes an angle θ with magnetic axis is given by \tt B_{e}=\frac{\mu_{0}}{4\pi}\frac{M}{d^3}\sqrt{1+3\cos^{2}\theta}
- If θ=0° \tt B=\frac{\mu_{0}}{4\pi}\ \frac{2M}{d^3}(Axial line)
- If If θ=90° \tt B=\frac{\mu_{0}}{4\pi}\ \frac{M}{d^3}(equatorial line)

### View the Topic in this video From 01:06 To 56:05

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1. The far axial magnetic field of a bar magnet which one may obtain experimentally. Thus, a bar magnet and a solenoid produce similar magnetic fields.

B = \frac{\mu_{0}}{4\pi} \frac{2m}{r^{3}}

2. The torque of the dipole in a magnetic field τ = m × B

3. In a simple hormonic motion, the square of angular frequency is and time period is

T = 2p \sqrt{\frac{I}{mB}} \ {\tt or} \ B = \frac{4 p^{2}I}{mT^{2}}

4. The Gauss's law of electrostatics, the flux through a closed surface in that case is given by

\sum E \cdot \Delta S = \frac{q}{\varepsilon_{0}}

5. Gauss's law of magnetism states that the flux of **B** through any closed surface is always zero \int_{s} **B**·d**s** = 0

6. Gauss's law of electrostatics, \int_{s} *B*·*ds* = μ_{0} *q _{m}* where

*q*is the magnetic charge

_{m}