Question

Two long, straight parallel wires are separated by a distance of \(20.0 \mathrm{~cm}\). Each wire carries a current of \(10.0 \mathrm{~A}\) in the same direction. What is the magnitude of the resulting magnetic field at a point that is \(12.0 \mathrm{~cm}\) from each wire?

Short answer

Question: Calculate the magnitude of the magnetic field at a point that is equidistant from two parallel wires carrying a current of \(10.0\,\mathrm{A}\) each, with one wire located \(12.0\,\mathrm{cm}\) from the point and the other wire \(20.0\,\mathrm{cm}\) away from it. Answer: To find the magnitude of the magnetic field at the given point, first calculate the magnetic fields produced by each wire separately using the Biot-Savart law. Then, use the Pythagorean theorem to find the total magnetic field at the point, since the magnetic fields from the two wires are perpendicular to each other. The resulting magnitude of the magnetic field at the given point can be found by performing these steps and calculations.

Step-by-step solution

Determine the B-field of each wire separately

We will use the Biot-Savart law, which gives us the formula for a magnetic B-field produced by a wire carrying current: $$B = \frac{\mu_0 I}{2 \pi r}$$ where \(B\) is the magnetic field, \(\mu_0\) is the permeability of free space (\(4\pi × 10^{-7} \mathrm{ T m / A}\)), \(I\) is the current and \(r\) is the distance from the wire.

Calculate the B-field of Wire 1

Using the information provided, the current and distance from the point for Wire 1 are \(I_1=10.0\,\mathrm{ A}\) and \(r_1=12.0\,\mathrm{ cm}\). Convert the distance to meters, i.e., \(r_1 = 0.12\,\mathrm{ m}\), and calculate the magnetic field \(B_1\) using the formula: $$B_1 = \frac{4\pi × 10^{-7} \mathrm{ T m / A} × 10.0\,\mathrm{ A}}{2 \pi × 0.12\,\mathrm{ m}}$$

Calculate the B-field of Wire 2

Now, the current and distance from the point for Wire 2 are \(I_2=10.0\,\mathrm{ A}\) and \(r_2=20.0\,\mathrm{ cm} - 12.0\,\mathrm{ cm} = 8.0\,\mathrm{ cm}\). Convert the distance to meters, i.e., \(r_2 = 0.08\,\mathrm{ m}\), and calculate the magnetic field \(B_2\) using the formula: $$B_2 = \frac{4\pi × 10^{-7} \mathrm{ T m / A} × 10.0\,\mathrm{ A}}{2 \pi × 0.08\,\mathrm{ m}}$$

Determine the directions of the B-fields

Since both currents are flowing in the same direction, the magnetic fields will have opposite directions due to the right-hand rule. At the given point, the magnetic fields from the two wires would intersect each other, forming a right angle.

Calculate the total magnetic field at the given point

To find the total magnetic field at the given point, we can use the Pythagorean theorem, since the two magnetic fields are perpendicular to each other: $$B_\text{total} = \sqrt{B_1^2 + B_2^2}$$ Calculate the values of \(B_1\) and \(B_2\) from Steps 2 and 3, and plug them into the formula above to find the total magnetic field at the given point. Upon performing these steps and calculations, the magnitude of the resulting magnetic field at a point that is \(12.0\,\mathrm{cm}\) from each wire will be found.