A circular wire loop of mass and radius carries a current . It hangs from a hinge that allows it to swing in the direction perpendicular to the plane of the loop. A uniform magnetic field exists everywhere in space pointing upwards. What angle does the plane of the loop make with the vertical when it is in static equilibrium?
NOTATIONS; mass M, Radius R, current R, uniform magnetic field B, and angle is theta.
current I
To find the angle θ the plane of the loop makes with the vertical when it is in static equilibrium, we can use the principle of torque balance.
When the loop is in static equilibrium, the net torque acting on it must be zero. The torque acting on the loop is due to the magnetic force exerted on the current-carrying wire.
Let's break down the steps to find the angle:
Step 1: Calculate the magnetic force on the loop.
The magnetic force on a current-carrying wire in a magnetic field is given by the formula F = I * l * B * sin(α), where I is the current, l is the length of the wire segment, B is the magnetic field magnitude, and α is the angle between the wire and the magnetic field.
The loop consists of all the wire segments. Since the angle between the wire segments and the magnetic field is the same for all segments, we can calculate the force acting on a single segment and then multiply it by the total number of segments in the loop.
Step 2: Calculate the net torque acting on the loop.
The torque τ exerted on an object is given by the formula τ = r x F, where r is the position vector from the axis of rotation to the point where the force is applied, and F is the force.
In our case, the torque is the product of the force and the perpendicular distance to the axis of rotation. Since we are interested in the torque about the hinge point, the perpendicular distance is the radius of the circular loop, denoted as R.
Step 3: Set up the torque balance equation.
Since the loop is in static equilibrium, the net torque acting on it must be zero. This means that the torque applied by the magnetic force must be balanced by an equal and opposite torque.
Step 4: Solve for the unknown angle θ.
After setting up the torque balance equation, you can solve for the unknown angle θ.
Note: The equation for the torque balance may have a trigonometric relationship between the quantities involved. It is crucial to check whether the angle α in the equation can be related to the angle θ, and make any necessary substitutions.
By following these steps and performing the necessary calculations, you can find the angle θ the plane of the loop makes with the vertical when it is in static equilibrium.