Consider a non - conducting disc or radius R and uniform surface charge density

Question

Consider a non - conducting disc or radius R and uniform surface charge density sigma. The disc rotates with an angular velocity omega. (a). Calculate the magnetic moment vector mu in terms of omega. (b). Recall from all the way back to last semester that tau = I alpha and that alpha = d omega/dt. If the mass of the disc is M_0, what is the magnitude of the constant torque necessary to bring this disc to a stop in a time T? (c). What is the external magnetic field B_ext, if any, that can bring about the torque from part (b)? Explain your reasoning ... an answer without an explanation will get zero credit.

Explanation / Answer

(a)

Let us consider a ring of radius r and width dr on the disc.

The charge dq on this ring is,

dq = 2rdr

The speed of charge here is v = R, so through a width dr on this ring charge dq passes in time 2/

So current I = dq/(2/) = 2rdr/(2/)

or, I = rdr

So magnetic moment is,

dµ = IA, where A is the area enclosed by the ring.

So, dµ = rdr(r2) = r3dr

So total magnetic moment can be found out by integrating the above equation and taking limit of r from 0 to R

So net magnetic moment is,

µ = R4/4, direction will be along .

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(b)

Moment of inertia of the disc is,

I0 = M0R2/2

Let the angular deceleration be , then to bring the disc stop in time T, we have

= T

or, = /T

or, Torque = I0 = I0/T

or, Torque = (M0R2/2)/T, direction of torque should be opposite to .

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(c)

Torque due to external field on magnetic moment is,

T = µXBext, so any torque produced due to external magnetic field will be perpendicular to both µ and Bext.

But we saw above in (b) that the torque needed to stop the disc must be in a direction opposite to , i.e., anti-parallel to . So, torque should also be anti-parallel to µ as both µ and have same direction.

But the torque produced by the external field will be perpendicular to µ.

So, there doesn't exist any external magnetic field that can produce the torque needed in (b).

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