A circular wire loop of mass M and radius R carries a current I . It hangs from a hinge that allows it to swing in the direction perpendicular to the plane of the loop. A uniform magnetic field B exists everywhere in space pointing upwards. What angle theta does the plane of the loop make with the vertical when it is in static equilibrium?
To determine the angle θ that the plane of the wire loop makes with the vertical in static equilibrium, we need to consider the forces acting on the loop.
1. Weight force (mg): This force acts vertically downward and can be calculated as mg = M * g, where M is the mass of the loop and g is the acceleration due to gravity.
2. Magnetic force (F_magnetic): The magnetic force acts perpendicular to both the current direction and the magnetic field. In this case, the current is in the plane of the loop, and the magnetic field is pointing upwards. The magnitude of the magnetic force can be calculated using the formula F_magnetic = I * L * B, where I is the current, L is the length of wire in the loop, and B is the magnetic field strength.
Now, let's consider the equilibrium condition. In static equilibrium, the sum of all forces acting on the loop must be equal to zero. Since the magnetic force is acting perpendicular to the plane of the loop, it will exert a torque about the axis of rotation (the hinge). This torque must be balanced by an equal and opposite torque to maintain equilibrium.
The torque due to the magnetic force can be calculated as τ = F_magnetic * R * sin(θ), where R is the radius of the loop and θ is the angle between the plane of the loop and the vertical.
The torque due to the weight force can be calculated as τ_weight = mg * R * sin(θ).
Setting these two torques equal, we can solve for θ:
F_magnetic * R * sin(θ) = mg * R * sin(θ)
Canceling out R and sin(θ) from both sides:
F_magnetic = mg
Now, substitute the values for the magnetic force and weight force:
I * L * B = M * g
Solving for θ:
θ = sin^(-1)((M * g) / (I * L * B))
Note: The above derivation assumes that the loop is fully vertical in the initial position, and it neglects any resistive or dissipative effects.