28.39 The figure shows a cross section across the diameter of a long, solid, cylindrical conductor. The radius of the cylinder is \(R=10.0 \mathrm{~cm} .\) A current of \(1.35 \mathrm{~A}\) is uniformly distributed through the conductor and is flowing out of the page. Calculate the direction and the magnitude of the magnetic field at positions \(r_{\mathrm{a}}=0.0 \mathrm{~cm}\) \(r_{\mathrm{b}}=4.00 \mathrm{~cm}, r_{\mathrm{c}}=10.00 \mathrm{~cm}\) and \(r_{d}=16.0 \mathrm{~cm} .\)

Question: Calculate and compare the magnetic fields at positions \(r_\text{a}=0.0 \mathrm{~cm},~ r_\text{b}=4.00 \mathrm{~cm},~ r_\text{c}=10.00 \mathrm{~cm}\) and \(r_\text{d}=16.0 \mathrm{~cm}\) in relation to the center of a cylindrical conductor with uniform current flowing out of the page. Use the current \(I = 1.35 \mathrm{~A}\) and the radius of the cylinder \(R = 10.0 \mathrm{~cm}\). Answer: The magnetic field at different positions are as follows: - At \(r_\text{a}\), the magnetic field is \(B_\text{a} = 0.0 \mathrm{~T}\). - At \(r_\text{b}\), the magnetic field is \(B_\text{b} = \frac{1}{2}\mu_{0}\left(\frac{1.35}{\pi(10)^{2}}\right)(4.00) \approx 5.42 \times 10^{-5} \mathrm{~T}\). - At \(r_\text{c}\), the magnetic field is \(B_\text{c} = \frac{1}{2}\mu_{0}\left(\frac{1.35}{\pi(10)^{2}}\right)(10.00) \approx 1.36 \times 10^{-4} \mathrm{~T}\). - At \(r_\text{d}\), the magnetic field is \(B_\text{d} = \frac{\mu_0 (1.35)}{2\pi(16.0)} \approx 5.37 \times 10^{-5} \mathrm{~T}\). The magnetic field is zero at the center of the conductor and increases as we move towards the surface. It reaches its maximum value on the surface and then decreases as we move away from the conductor. The magnetic field direction at each position is tangent to the circular loop, following the right-hand rule.