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 θ does the plane of the loop make with the vertical when it is in static equilibrium?
To find the angle θ at which the plane of the loop makes with the vertical when it is in static equilibrium, we need to consider the forces acting on the loop.
1. Gravitational force: The loop is hanging from a hinge, so there is a downward gravitational force acting on it. This force can be calculated as Fg = Mg, where M is the mass of the loop and g is the acceleration due to gravity.
2. Magnetic force: The current carrying loop experiences a force due to the interaction with the magnetic field. This force is given by Fm = I * B * A * sin(α), where I is the current, B is the magnetic field strength, A is the area of the loop, and α is the angle between the magnetic field and the plane of the loop.
The area of the loop is given by A = π * R^2, where R is the radius of the loop.
3. Tension force: Since the loop is in static equilibrium, the tension in the wire must balance the gravitational and magnetic forces. The tension force can be calculated as Ft = Fg + Fm.
Now, at the static equilibrium, the tension force Ft must act along the vertical direction. To achieve this, the magnetic force Fm and the gravitational force Fg must have equal components in the vertical direction.
The component of the magnetic force in the vertical direction is Fm * sin(α), and the component of the gravitational force in the vertical direction is Fg * cos(θ), where θ is the angle made by the plane of the loop with the vertical.
Equating these two components, we have Fg * cos(θ) = Fm * sin(α). Substituting the expressions for Fg and Fm, we get Mg * cos(θ) = I * B * A * sin(α).
Substituting the expressions for A and α, we have Mg * cos(θ) = I * B * π * R^2 * sin(α).
Now, we can solve this equation for θ.
Note: The direction of the magnetic force is given by the right-hand rule, where the thumb points in the direction of the current and the curled fingers represent the magnetic field.
I hope this explanation helps you understand the steps needed to find the angle θ in the given scenario. If you have the specific values for the mass M, radius R, current I, and magnetic field strength B, you can plug in those values into the equation to find the angle θ.