derive an expression for period of a magnet kept in uniform magnetic field
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Answer:
In this section we will learn about the nature of forces acting on a dipole placed in a uniform magnetic field and will compare it with the case when a dipole is kept in an electrostatic field. As we know that if we place iron fillings around a bar magnet on a sheet of paper and tap the sheet, the fillings rearrange themselves to form a specific pattern. The pattern of iron filings here denote the magnetic field lines generated due to the magnet. These magnetic field lines give us an approximate idea of the magnetic field B. But many a times we are required to determine the magnitude of the magnetic field B accurately. We accomplish this by placing a small compass needle of known magnetic moment m and moment of inertia and allow it to oscillate in the magnetic field.
The torque on the needle can be given as,
τ=m×B
The magnitude of this torque is given by mB sinƟ. Here τ is the restoring torque, and Ɵ is the angle between the direction of the magnetic moment (m) and the direction of the magnetic field (B).
At equilibrium, we can say that,
Id2θdt≅−mBsinθ
The negative sign in the above expression mB sinθ brings us to the conclusion that the restoring torque acting here acts in the opposite direction to the deflecting torque. Also, as the value of θ is very small in radians, we can approximate sin θ ≈ θ. Thus, using this approximation, we can write
Id2θdt≅−mBθ
Or,
d2θdt≅−mBθI
The above equation is a representation of a simple harmonic motion and the angular frequency can be given as,
ω=mBI
and thus, the time period can be stated as,
T=2πImB−−−√
Or, we can also write it as,
B=4π2ImT2
The expression for magnetic potential energy is derived in the same manner as we derive the electrostatic potential energy as can be seen below. The magnetic potential energy Um can be given by
Um=∫τ(θ)dθ∫mBsinθ=−mBcosθ=−m.B
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