Question

Two 50 -turn coils, each with a diameter of \(4.00 \mathrm{~m}\), are placed \(1.00 \mathrm{~m}\) apart, as shown in the figure. \(\mathrm{A}\) current of \(7.00 \mathrm{~A}\) is flowing in the wires of both coils; the direction of the current is clockwise for both coils when viewed from the left. What is the magnitude of the magnetic field in the center between the two coils?

Short answer

Answer: The magnitude of the magnetic field in the center between the two coils is approximately \(2.48 \times 10^{-5}\,\text{T}\).

Step-by-step solution

Calculate Magnetic Field due to a Single Coil

To find the magnetic field due to a single coil, we will use the formula for magnetic field created by a current-carrying wire loop at a point on its axis: $$B = \frac{\mu_0NI}{2R}\frac{x}{(x^2 + R^2)^{3/2}}$$ In this formula, \(\mu_0\) is the permeability of free space (\(4 \pi \times 10^{-7} \,\text{T} \cdot \text{m} / \text{A}\)), \(N\) is the number of turns in the coil, \(I\) is the current, \(R\) is the coil radius, and \(x\) is the distance from the coil center to the point where we want to calculate the magnetic field. For our problem, we will calculate the magnetic fields due to each coil at the center point between the two coils. Therefore, we need to find the magnetic fields at \(x = R/2\) for each coil.

Calculate the Radius and Distance

Given that the diameter of the coils is \(4.00\,\text{m}\), the radius of each coil is: $$R = \frac{d}{2} = \frac{4.00\,\text{m}}{2} = 2.00\,\text{m}$$ The distance from the coil center to the point between the two coils is half the distance between the coils: $$x = \frac{1.00\,\text{m}}{2} = 0.50\,\text{m}$$

Calculate the Magnetic Field for each Coil

Now, we can plug in the values into the formula for the magnetic field: $$B_\text{coil} = \frac{(4 \pi \times 10^{-7}\, \text{T}\cdot \text{m} / \text{A})(50)(7.00\,\text{A})}{2(2.00\,\text{m})}\frac{0.50\,\text{m}}{(0.50^2 + 2.00^2)^{3/2}}$$ By evaluating these values, we will get the magnetic field due to one coil at the point between the coils.

Calculate the Total Magnetic Field

Since both coils have the same dimensions and current flows in the same direction, their magnetic fields will add up in the center point between the two coils. So, we will multiply the magnetic field due to a single coil by \(2\) to get the total magnetic field: $$B_\text{total} = 2B_\text{coil}$$

Calculate and Present the Result

Now, we can calculate the total magnetic field and present the result: $$B_\text{total} = 2B_\text{coil} \approx 2(1.24 \times 10^{-5}\,\text{T}) = 2.48 \times 10^{-5}\,\text{T}$$ So, the magnitude of the magnetic field in the center between the two coils is approximately \(2.48 \times 10^{-5}\,\text{T}\).