Question

A straight wire with a constant current running through it is in Earth's magnetic field, at a location where the magnitude is \(0.430 \mathrm{G}\). What is the minimum current that must flow through the wire for a \(10.0-\mathrm{cm}\) length of it to experience a force of \(1.00 \mathrm{~N}\) ?

Short answer

Answer: The minimum current required is approximately \(2.33\times10^4\,\mathrm{A}\).

Step-by-step solution

Convert given data to appropriate units

We have the magnetic field given as \(0.430\,\mathrm{G}\), which should be converted to tesla (T). The conversion factor is \(1\,\mathrm{T}=10^4 \,\mathrm{G}\). Also, the length of the wire is given in centimeters, so we should convert it to meters. \(B = 0.430 \times 10^{-4}\,\mathrm{T}\) \(L = 10.0\,\mathrm{cm} \times \frac{1\,\mathrm{m}}{100\,\mathrm{cm}} = 0.100\,\mathrm{m}\)

Rearrange the equation for magnetic force

Our goal is to find the current \(I\). We can rearrange the formula for the magnetic force to solve for \(I\): \(F = BIL\sin\theta\) Since we want the minimum current for the maximum force, we can assume that \(\sin\theta = 1\), or \(\theta = 90°\). So the formula becomes: \(I = \frac{F}{BL}\)

Substitute the values and find the current

Now we can plug in the values for the force \(F\), the magnetic field strength \(B\), and the wire length \(L\): \(I = \frac{1.00\,\mathrm{N}}{(0.430 \times 10^{-4}\,\mathrm{T})(0.100\,\mathrm{m})}\) Calculating the expression gives: \(I = \frac{1.00\,\mathrm{N}}{0.430\times10^{-5}\,\mathrm{T}\cdot\mathrm{m}} \approx 2.33\cdot10^4\,\mathrm{A}\) The minimum current that must flow through the wire for a \(10.0\,\mathrm{cm}\) length of it to experience a force of \(1.00\,\mathrm{N}\) is approximately \(2.33\times10^4\,\mathrm{A}\).