Magnetism and Matter

5.6 Magnetic field strength, B = 0.25 T

Magnetic moment, M = 0.6 J/T

The angle $\theta$ , between the axis of the solenoid and the direction of the applied field is 30 $°$ . Therefore, the torque acting on the solenoid is given as

$\tau =MBsin\theta$ = 0.6 $*0.25*\sin ?30°$ = 7.5 $*{10}^{-2}$ J

5.5 Number of turns, n = 800

Area of the cross-section, A = 2.5 $*{10}^{-4}$ $m^2$

Current flowing, I = 3.0 A

A current carrying solenoid behaves like a bar magnet because a magnetic field develops along its axis (along the length). $°$

The magnetic moment associated is calculated as M = nIA = 800 $*3*$ 2.5 $*{10}^{-4}$ J/T= 0.6 J/T

5.4 Moment of the bar magnet, M = 0.32 J/T

Magnetic field, B = 0.15 T

The bar magnet is aligned along the magnetic field. This system is considered as being in stable equilibrium. Hence, the angle $\theta$ , between the bar magnet and the magnetic field is 0 $°$ .

Potential energy of the system = -MBcos $\theta$ = - 0.32 $*0.15*\cos ?0°$ = -4.8 $*{10}^{-2}$ J

When the bar magnet is oriented 180 $°$ to the magnetic field, it becomes unstable equilibrium.

Potential energy = - MBcos $\theta$ = - 0.32 $*0.15*\cos ?180°$ = 4.8 $*{10}^{-2}$ J

5.3 Magnetic field strength, B = 0.25 T

Torque on the bar magnet, $\tau$ = 4.5 $*{10}^{-2}$ J

Angle between the bar magnet and the external magnetic field, $\theta$ = 30 $°$

From the relation T = MB $\sin ?\theta$ , where M = Magnetic moment, we get

M = $\frac{\tau }{B\sin ?\theta }$ = $\frac{4.5\mathrm{}*{10}^{-2}}{0.25sin30°}$ = 0.36 J/T

Hence the magnetic moment is 0.36 J/T

The net magnetic flux through any closed surface is zero, according to Gauss's Law for magnetism.
Mathematically,
? B · dA = 0.
It states that all magnetic field lines that enter a surface must exit from it as well. In this way, it is different from the electric charges which can exist independently as positive and negative.

The Earth's magnetic field makes a magnetic inclination or dip with the horizontal plane at a particular location. It shows how much the magnetic field is tilted from the horizontal. It gets tilted due to the Earth's magnetic nature. The formula is:
tan I = B_V / B_H,
where B_V is the vertical component and B_H is the horizontal component of the Earth's magnetic field.

The angle between the magnetic north (the direction a compass points) and the geographic north (true north) is the magnetic declination. This angle can change over time due to changes in the Earth's magnetic field and varies with location on Earth. It is important for navigation where accurate directional orientation is crucial, especially in marine and aviation transport. The declination helps in avoiding navigational errors as it ensures that compass readings are corrected.

The Earth's magnetic field makes a magnetic inclination or dip with the horizontal plane at a particular location. It shows how much the magnetic field is tilted from the horizontal. It gets tilted due to the Earth's magnetic nature. The formula is:
tan I = B_V / B_H,
where B_V is the vertical component and B_H is the horizontal component of the Earth's magnetic field.