## Question

The 20.0 cm×35.0 cm rectangular circuit shown in the figure is hinged along side ab. It carries a clockwise 5.00-A current and is located in a uniform 1.20-T magnetic field oriented perpendicular to two of its sides, as shown. (a) Draw a clear diagram showing the direction of the force that the magnetic field exerts on each segment of the circuit ( ab,bcr, etc.). (b) Of the four forces you drew in part (a), decide which ones exert a torque about the hinge ob. Then calculate only those forces that exert this torque. (c) Use your results from part (b) to calculate the torque that the magnetic field exerts on the circuit about the hinge axis ab.

## Answer

Solution to Magnetic Field Torque on a Rectangular Circuit

**a) Direction of magnetic forces:**

**Segment ab:**Since the current flows inward on this side and the magnetic field points downward, the magnetic force will point outward (to the right) on ab.**Segment bc:**The current flows downward on bc, and the magnetic field points downward. Therefore, the magnetic force will be to the left on bc.**Segment cd:**Here, the current flows to the left, and the magnetic field points downward. Thus, the magnetic force will be upward on cd.**Segment da:**Finally, the current flows upward on da, and the magnetic field points downward. This results in a magnetic force directed downward on da.

**b) Forces creating torque:**

Only the forces acting perpendicular to the hinge axis (ab) contribute to the torque. Hence, the forces on da and bc do not produce any torque. Therefore, we need to calculate only the forces on ab and cd.

**c) Calculating torque:**

**Force on ab:**As mentioned in part a), the force on ab points to the right. Its magnitude is given by F_ab = Iab * B, where Iab = 5.00 A is the current in segment ab and B = 1.20 T is the magnetic field strength. Therefore, F_ab = 5.00 A * 1.20 T = 6.00 N.**Distance from ab to hinge (perpendicular):**This distance is the height of the rectangle, 20.0 cm = 0.20 m.**Force on cd:**The force on cd points upward with a magnitude F_cd = Icd * B = 5.00 A * 1.20 T = 6.00 N.**Distance from cd to hinge (perpendicular):**This distance is the width of the rectangle, 35.0 cm = 0.35 m.

**Torque calculation:**

The torque about the hinge axis is the sum of the moments of the forces acting perpendicular to the hinge. Hence,

```
τ = F_ab * d_ab - F_cd * d_cd
= 6.00 N * 0.20 m - 6.00 N * 0.35 m
= -0.60 N·m
```

Therefore, the torque exerted by the magnetic field on the circuit about the hinge axis ab is **-0.60 N·m**, acting clockwise (negative sign).

**Note:** The negative sign indicates that the torque tends to rotate the circuit in a counter-clockwise direction, opposing the initial clockwise current flow.