Let's solve the problem step by step: ### **Given Data** - Magnetic field: \(\mathbf{H} = \rho z^2 \, \mathbf{a}_\rho\) A/m - Surface: \(\rho = 2\), \(0 < z < 2\), \(0 < \phi < \frac{\pi}{2}\) - Region: Free space ### **Step 1: Magnetic Flux Definition** The **magnetic flux** through a surface \(S\) is: \[ \Psi_m = \int_S \mathbf{B} \cdot d\mathbf{A} \] But in free space, \(\mathbf{B} = \mu_0 \mathbf{H}\). So, \[ \Psi_m = \int_S \mu_0 \mathbf{H} \cdot d\mathbf{A} \] ### **Step 2: Surface Description** - The surface is defined by \(\rho = 2\) (a cylindrical surface), and it extends over \(0 < z < 2\) and \(0 < \phi < \frac{\pi}{2}\). - The **differential area vector** for a surface of constant \(\rho\) in cylindrical coordinates is: \[ d\mathbf{A} = (\rho \, d\phi \, dz) \mathbf{a}_\rho \] (The area vector points radially outward.) ### **Step 3: Setting Up the Integral** On the surface \(\rho = 2\): - \(\mathbf{H} = 2z^2 \, \mathbf{a}_\rho\) - \(d\mathbf{A} = (2 \, d\phi \, dz) \mathbf{a}_\rho\) So, \[ \mathbf{H} \cdot d\mathbf{A} = (2z^2) \cdot (2 \, d\phi \, dz) = 4z^2 \, d\phi \, dz \] Total flux: \[ \Psi_m = \mu_0 \int_{z=0}^{2} \int_{\phi=0}^{\pi/2} 4z^2 \, d\phi \, dz \] ### **Step 4: Evaluate the Integrals** First, integrate over \(\phi\): \[ \int_{\phi=0}^{\pi/2} d\phi = \frac{\pi}{2} \] So, \[ \Psi_m = \mu_0 \int_{z=0}^{2} 4z^2 \left(\frac{\pi}{2}\right) dz = 2\pi \mu_0 \int_{z=0}^2 z^2 dz \] Now, integrate \(z^2\): \[ \int_{z=0}^2 z^2 dz = \left[ \frac{z^3}{3} \right]_{0}^{2} = \frac{8}{3} \] So, \[ \Psi_m = 2\pi \mu_0 \times \frac{8}{3} = \frac{16\pi}{3} \mu_0 \] ### **Step 5: Final Answer** \[ \boxed{\Psi_m = \frac{16\pi}{3} \mu_0} \] where \(\mu_0 = 4\pi \times 10^{-7} \, \text{H/m}\) (if a numerical value is needed). --- **Summary of Steps:** 1. Wrote the magnetic flux formula for free space. 2. Found the differential area vector for the given surface. 3. Set up the double integral over the surface. 4. Evaluated the integrals step by step. 5. Provided the final boxed answer.

Certainly! Here are the step-by-step explanations, each summarized in approximately 50 words: **Step 1: Magnetic Flux Definition** Magnetic flux through a surface is the surface integral of the magnetic flux density \(\mathbf{B}\). Since \(\mathbf{B} = \mu_0 \mathbf{H}\) in free space, the flux can be calculated by integrating \(\mu_0 \mathbf{H}\) over the given surface, considering the orientation and area vector. **Step 2: Surface Description** The surface is a cylindrical segment at \(\rho=2\). Its differential area element in cylindrical coordinates is aligned with the radial direction (\(\mathbf{a}_\rho\)). The area vector points outward radially, and the integration limits are set for the specified \(\phi\), \(z\), and \(\rho\). **Step 3: Setting Up the Integral** Express \(\mathbf{H}\) at \(\rho=2\), which simplifies to \(2z^2 \mathbf{a}_\rho\). The differential area vector \(d\mathbf{A}\) points radially outward with magnitude \(2\,d\phi\,dz\). The dot product yields the integrand for flux calculation. **Step 4: Evaluate the Integrals** Perform the \(\phi\)-integration over \(0\) to \(\pi/2\), giving \(\pi/2\). Then, integrate \(z^2\) from 0 to 2, resulting in \(8/3\). Multiply these results by constants to find the total flux expression, simplifying step by step. **Step 5: Final Answer** Combine the results, including \(\mu_0\), to obtain the total magnetic flux through the surface. The final expression is \(\displaystyle \frac{16\pi}{3} \mu_0\), representing the flux in terms of fundamental constants and the geometry of the problem.