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Chapter 27: Problem 36

A straight wire of length \(2.00 \mathrm{~m}\) carries a current of \(24.0 \mathrm{~A} .\) It is placed on a horizontal tabletop in a uniform horizontal magnetic field. The wire makes an angle of \(30.0^{\circ}\) with the magnetic field lines. If the magnitude of the force on the wire is \(0.500 \mathrm{~N}\), what is the magnitude of the magnetic field?

Short Answer

Expert verified
Answer: The magnitude of the magnetic field is approximately 0.0208 T (tesla).

Step by step solution

01

Write down the known values

We are given the following values: - Length of the wire (\(L\)) = 2.00 m - Current in the wire (\(I\)) = 24.0 A - Angle between the wire and magnetic field lines (\(\theta\)) = 30.0° - Force on the wire (\(F\)) = 0.500 N
02

Convert the angle to radians

In order to use the formula, we need to convert the angle from degrees to radians. We can do this using the following conversion: 1 radian = 180°/π. So, we have: $$ \theta_{rad} = 30.0° \cdot \frac{\pi}{180°} = \frac{\pi}{6} \, \text{radians} $$
03

Use the formula

Now that we have all the necessary values, we can use the formula for the force on a wire in a magnetic field, \(F = BIL\sin{\theta}\), to find the magnitude of the magnetic field, \(B\): $$ B = \frac{F}{IL\sin{\theta}} $$
04

Substitute the known values

Now, substitute the known values into the formula: $$ B = \frac{0.500 N}{(24.0 A)(2.00 m) \sin(\frac{\pi}{6})} $$
05

Calculate the magnetic field magnitude

Finally, calculate the magnetic field magnitude: $$ B = \frac{0.500 N}{(24.0 A)(2.00 m) \, (0.5)} = \frac{0.500 N}{24.0 A} $$ The magnitude of the magnetic field is approximately: $$ B \approx 0.0208 \, \text{T} $$ So the magnitude of the magnetic field is approximately 0.0208 T (tesla).

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Most popular questions from this chapter

A high electron mobility transistor (HEMT) controls large currents by applying a small voltage to a thin sheet of electrons. The density and mobility of the electrons in the sheet are critical for the operation of the HEMT. HEMTs consisting of AlGaN/GaN/Si are being studied because they promise better performance at higher powers, temperatures, and frequencies than conventional silicon HEMTs can achieve. In one study, the Hall effect was used to measure the density of electrons in one of these new HEMTs. When a current of \(10.0 \mu\) A flows through the length of the electron sheet, which is \(1.00 \mathrm{~mm}\) long, \(0.300 \mathrm{~mm}\) wide, and \(10.0 \mathrm{nm}\) thick, a magnetic field of \(1.00 \mathrm{~T}\) perpendicular to the sheet produces a voltage of \(0.680 \mathrm{mV}\) across the width of the sheet. What is the density of electrons in the sheet?

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