Question

A copper sheet with length \(1.00 \mathrm{~m}\), width \(0.500 \mathrm{~m},\) and thickness \(1.00 \mathrm{~mm}\) is oriented so that its largest surface area is perpendicular to a magnetic field of strength \(5.00 \mathrm{~T}\). The sheet carries a current of \(3.00 \mathrm{~A}\) across its length. What is the magnitude of the force on this sheet? How does this magnitude compare to that of the force on a thin copper wire carrying the same current and oriented perpendicularly to the same magnetic field?

Short answer

Answer: The magnitudes of the magnetic forces on the copper sheet and the thin copper wire are both 15.0 N.

Step-by-step solution

1. Calculate the magnetic force on the copper sheet

: We are given the dimensions of the copper sheet and its orientation with respect to the magnetic field. The magnetic force on a current-carrying conductor can be calculated using the formula: \(F = BIL\) where \(F\) is the magnetic force, \(B\) is the magnetic field strength, \(I\) is the current, and \(L\) is the length of the conductor. In this case, we have: \(B = 5.00 \ \text{T}\) (magnetic field strength), \(L = 1.00 \ \text{m}\) (length of the conductor), \(I = 3.00 \ \text{A}\) (current across the length of the sheet). Now, calculate the magnetic force: \(F_{sheet} = BIL = 5.00 \times 3.00 \times 1.00 = 15.0 \ \text{N}\) The magnitude of the force on the copper sheet is \(15.0 \ \text{N}\).

2. Calculate the magnetic force on the thin copper wire

: Since there is no information about the length or thickness of the copper wire, we should assume that the orientation of the wire and the sheet is the same in the magnetic field. This means that the copper wire will experience the same magnetic force as the copper sheet because the formula for calculating the magnetic force considers only the current and the length of the conductor, both of which are the same in this case. Therefore, the force on the copper wire carrying the same current and oriented perpendicularly to the same magnetic field will be the same as the magnetic force on the copper sheet: \(F_{wire} = F_{sheet} = 15.0 \ \text{N}\)

3. Compare the magnitude of the forces

: The magnetic force on the thin copper wire carrying the same current and oriented perpendicularly to the same magnetic field is equal to the force on the copper sheet: \(F_{wire} = F_{sheet} = 15.0 \ \text{N}\) This implies that the magnitudes of the forces are the same for the copper sheet and the thin copper wire in this scenario.