Problem 7 Solution
1. Physics Principle:
A superconductor expels magnetic field lines (Meissner effect). When placed in a magnetic field, surface currents are induced to cancel the field inside. This interaction results in a magnetic pressure on the surface, creating an upward repulsive force.
The magnetic pressure $P_m$ is proportional to the square of the magnetic induction: $$ P_m \propto B^2 $$ Consequently, the upward magnetic force $F_m$ is proportional to $B^2$: $$ F_m = k B^2 $$ where $k$ is a constant depending on the geometry of the cube.
A superconductor expels magnetic field lines (Meissner effect). When placed in a magnetic field, surface currents are induced to cancel the field inside. This interaction results in a magnetic pressure on the surface, creating an upward repulsive force.
The magnetic pressure $P_m$ is proportional to the square of the magnetic induction: $$ P_m \propto B^2 $$ Consequently, the upward magnetic force $F_m$ is proportional to $B^2$: $$ F_m = k B^2 $$ where $k$ is a constant depending on the geometry of the cube.
2. Case 1: Unloaded Cube
The superconducting cube of mass $m$ hovers. The upward magnetic force balances gravity: $$ F_{m1} = mg $$ $$ k B_1^2 = mg \quad \dots(1) $$
The superconducting cube of mass $m$ hovers. The upward magnetic force balances gravity: $$ F_{m1} = mg $$ $$ k B_1^2 = mg \quad \dots(1) $$
3. Case 2: Loaded Cube
A non-magnetic cube of the same mass $m$ is placed on top. The total mass is now $2m$. To make this system hover, the magnetic field is increased to $B_2$. $$ F_{m2} = 2mg $$ $$ k B_2^2 = 2mg \quad \dots(2) $$
A non-magnetic cube of the same mass $m$ is placed on top. The total mass is now $2m$. To make this system hover, the magnetic field is increased to $B_2$. $$ F_{m2} = 2mg $$ $$ k B_2^2 = 2mg \quad \dots(2) $$
4. Calculation:
Dividing equation (2) by equation (1): $$ \frac{k B_2^2}{k B_1^2} = \frac{2mg}{mg} $$ $$ \frac{B_2^2}{B_1^2} = 2 $$ $$ B_2 = \sqrt{2} B_1 $$
Dividing equation (2) by equation (1): $$ \frac{k B_2^2}{k B_1^2} = \frac{2mg}{mg} $$ $$ \frac{B_2^2}{B_1^2} = 2 $$ $$ B_2 = \sqrt{2} B_1 $$
Final Answer:
The magnetic induction must be increased by a factor of $\sqrt{2}$.
The magnetic induction must be increased by a factor of $\sqrt{2}$.