Question

A straight wire segment of length \(25 \mathrm{cm}\) carries a current of $33.0 \mathrm{A}$ and is immersed in a uniform external magnetic field. The magnetic force on the wire segment has magnitude \(4.12 \mathrm{N} .\) (a) What is the minimum possible magnitude of the magnetic field? (b) Explain why the given information enables you to calculate only the minimum possible field strength.

Short answer

Answer: The minimum possible magnetic field strength is 0.0499 T.

Step-by-step solution

Recall the formula for magnetic force

The formula for magnetic force on a current-carrying wire in a magnetic field is given by: \(F = BIL\sin\theta\), where \(F\) is the magnetic force, \(B\) is the magnetic field strength, \(I\) is the current, \(L\) is the length of the wire, and \(\theta\) is the angle between the direction of the current and the magnetic field.

Identify the given values and what needs to be found

We are given the values for the length of the wire \(L = 25\,\text{cm} = 0.25\,\text{m}\), the current \(I = 33.0\,\text{A}\), and the force \(F = 4.12\,\text{N}\). Our goal is to find the minimum possible magnetic field strength \(B_{\text{min}}\).

Find the condition for minimum magnetic field strength

The magnetic field strength will be minimum when the angle \(\theta\) between the direction of the current and the magnetic field is \(90^{\circ}\), as the sine function becomes maximum (\(\sin90^{\circ} = 1\)). In this case, \(F = B_{\text{min}}IL\).

Calculate the minimum magnetic field strength

Using the formula mentioned in the previous step, we can find \(B_{\text{min}}\): \(B_{\text{min}} = \frac{F}{IL} = \frac{4.12\,\text{N}}{(33.0\,\text{A})(0.25\,\text{m})} = 0.0499\,\text{T}\)

Explanation for why the given information enables to calculate only the minimum possible field strength

The minimum possible field strength is calculated based on the condition that the angle \(\theta\) between the current and the magnetic field is \(90^{\circ}\), which maximizes the force acting on the wire. However, we do not have information about the actual angle \(\theta\) in the problem, and therefore cannot calculate a specific field strength if \(\theta\) is not \(90^{\circ}\). The problem only provides enough information to find the minimum possible field strength, which is when the force is maximized and the sine component is at its maximum value (1).