Question

A \(20.0 \mathrm{cm} \times 30.0 \mathrm{cm}\) rectangular loop of wire carries \(1.0 \mathrm{A}\) of current clockwise around the loop. (a) Find the magnetic force on each side of the loop if the magnetic ficld is 2.5 T out of the page. (b) What is the net magnetic force on the loop?

Short answer

Answer: The magnetic force on each side parallel to the magnetic field is 0 N, the magnetic force on each side perpendicular to the magnetic field is 0.75 N, and the net magnetic force on the loop is 1.5 N.

Step-by-step solution

Setting up the problem

Determine the lengths of the sides of the rectangular loop. Using the given dimensions, we know that the loop has two sides with lengths of 20.0 cm and two sides with lengths of 30.0 cm.

Calculate the magnetic force on the sides parallel to the magnetic field

Using the formula \(F = I\mathcal{L} \times B\), we calculate the magnetic force on the two sides with lengths of 20.0 cm: \(F_{1} = I\mathcal{L}_{1} \times B = (1.0\,\mathrm{A})(0.20\,\mathrm{m})(2.5\,\mathrm{T}) = 0.5\,\mathrm{N}\). For these sides, since the current is parallel to the magnetic field, there is no magnetic force acting on them, so \(F_{1} = 0\).

Calculate the magnetic force on the sides perpendicular to the magnetic field

Now, we calculate the magnetic force on the two sides with lengths of 30.0 cm: \(F_{2} = I\mathcal{L}_{2} \times B = (1.0\,\mathrm{A})(0.30\,\mathrm{m})(2.5\,\mathrm{T}) = 0.75\,\mathrm{N}\). For these sides, the current is perpendicular to the magnetic field, so the magnetic force is maximized.

Calculate the net magnetic force

We now have the magnetic forces for each of the sides: \(F_{1} = 0\) and \(F_{2} = 0.75\,\mathrm{N}\). Since the forces on the two sides parallel to the magnetic field are zero, the net magnetic force is the sum of the forces on the other two sides: \(F_{\mathrm{net}} = 2F_{2} = 2(0.75\,\mathrm{N}) = 1.5\,\mathrm{N}\). So, the magnetic force on each side parallel to the magnetic field is 0 N, the magnetic force on each side perpendicular to the magnetic field is 0.75 N, and the net magnetic force on the loop is 1.5 N.