A thin aluminum ring hangs vertically from a torsion spring. A torsion spring when twisted exerts a restoring torque given by
τ=−κθ
where θ is the angle of twist. Suppose that the ring undergoes small torsional oscillations while it is being cooled down to the point where it becomes superconducting. The period of torsional oscillations of the superconducting ring is T0. This period changes after applying an external horizontal magnetic field of induction B parallel to the plane of the ring corresponding to θ=0 (the position of equilibrium). Show that for the case of a weak magnetic field B, the new period of oscillations is
T=T(0)−ΔT withΔT=Ca^4T(0)^3B^2/(JL).
Here, a is the radius of the cold ring, J is the moment of inertia with respect to the vertical axis (J=1/2ma^2), L is the ring's self inductance and C is a numerical coefficient. Determine the coefficient C.
Details and assumptions
Hint: (1+x)^α≈1+αx for x≪1.
To find the coefficient C, we need to use the given information and make use of the approximations mentioned in the hint.
Let's start by examining the equation for the new period of oscillations, T.
T = T(0) - ΔT
where ΔT is the change in the period due to the applied magnetic field.
The expression for ΔT is given as:
ΔT = Ca^4T(0)^3B^2 / (JL)
To determine C, we can manipulate the expression for ΔT by incorporating the given hint's approximation:
(1 + x)^α ≈ 1 + αx (for x ≪ 1)
Now, let's substitute some values into the equation for ΔT:
ΔT = Ca^4T(0)^3B^2 / (JL)
ΔT = C(1) * a^4 * (T(0)^3) * (B^2 / JL)
Since B is a weak magnetic field, we can assume it is small (B ≪ 1), which allows us to use the hint's approximation. We will approximate B^2 to 0 in the expansion.
So, let's rewrite the equation for ΔT incorporating this approximation:
ΔT ≈ C * a^4 * (T(0)^3) * (0 / JL)
ΔT ≈ 0
From this approximation, we can deduce that the change in the period ΔT is approximately 0 for weak magnetic fields.
Now, let's reconsider the equation for the new period T:
T = T(0) - ΔT
T = T(0) - 0
T = T(0)
From this, we can conclude that the new period T is equal to the initial period T(0) and does not change due to the weak magnetic field.
Since ΔT is 0, it implies that C is also 0. Therefore, the coefficient C is equal to zero.
So, the final result is that the coefficient C is zero, meaning there is no change in the period of oscillations due to the weak magnetic field.