Question

A coil consists of \(120 .\) circular loops of wire of radius \(4.80 \mathrm{~cm} .\) A current of 0.490 A runs through the coil, which is oriented vertically and is free to rotate about a vertical axis (parallel to the \(z\) -axis). It experiences a uniform horizontal magnetic field in the positive \(x\) -direction. When the coil is oriented parallel to the \(x\) -axis, a force of \(1.20 \mathrm{~N}\) applied to the edge of the coil in the positive \(y\) -direction can keep it from rotating. Calculate the strength of the magnetic field.

Short answer

Answer: The strength of the magnetic field is approximately \(7.13 \times 10^{-3}\,\text{T}\).

Step-by-step solution

Calculate the magnetic force on the coil due to the magnetic field

To find the strength of the magnetic field, we'll need the magnetic force acting on the coil due to the magnetic field. The formula for the magnetic force on a current-carrying loop is given by: \(F_B = N \cdot I \cdot A \cdot B \cdot sin \theta\) Where: F_B = magnetic force on the coil N = number of loops in the coil (120 loops) I = current in the coil (0.490A) A = area of each loop (\(pi r^2\), with r = 0.048m) B = strength of the magnetic field sinθ = sine of the angle between the normal to the plane of the coil and the direction of the magnetic field (1 for parallel orientation)

Calculate the torque on the coil due to the applied force

The next step is to determine the torque on the coil due to the applied force. We can calculate the torque using the formula: \(τ = F \cdot d\) Where: τ = torque F = applied force (1.20 N) d = distance from the axis of rotation to the point where the force is applied (radius of the coil, 0.048m)

Calculate the torque on the coil due to the magnetic field

Since the coil is not rotating, the torque due to the applied force is equal to the torque due to the magnetic force. We can calculate the torque due to the magnetic force using the formula: \(τ = N \cdot I \cdot A \cdot B \cdot r\) Where: τ = torque N, I, A, and r are the same as from Step 1 B = strength of the magnetic field

Find the strength of the magnetic field

Now that we have the torque due to the applied force and the magnetic force, we can equate them to find the strength of the magnetic field: \(1.20\cdot0.048 = 120 \cdot 0.490 \cdot (\pi\cdot0.048^2) \cdot B\) Solving for B, we get: \(B = \frac{1.20\cdot0.048}{120 \cdot 0.490 \cdot (\pi\cdot0.048^2)}\) Calculating the value of B: \(B \approx 7.13 \times 10^{-3}\,\text{T}\) Hence, the strength of the magnetic field is approximately \(7.13 \times 10^{-3}\,\text{T}\).