Given the field \(\mathbf{H}=20 \rho^{2} \mathbf{a}_{\phi} \mathrm{A} / \mathrm{m}:(a)\) Determine the current density \(\mathbf{J}\). (b) Integrate \(\mathbf{J}\) over the circular surface \(\rho \leq 1,0<\phi<2 \pi, z=0\), to determine the total current passing through that surface in the \(\mathbf{a}_{z}\) direction. (c) Find the total current once more, this time by a line integral around the circular path \(\rho=1,0<\phi<2 \pi, z=0 .\)

We have the magnetic field \(\mathbf{H}=20 \rho^{2} \mathbf{a}_{\phi} \mathrm{A} /\mathrm{m}\), and we need to find the current density \(\mathbf{J}\). From Ampere's law, we know that \(\nabla \times \mathbf{H} = \mathbf{J}\). Using the formula for curl in cylindrical coordinates, we have: $$ \mathbf{J} = \nabla \times \mathbf{H} = \left( \frac{1}{\rho} \frac{\partial H_z}{\partial \phi} - \frac{\partial H_\phi}{\partial z} \right) \mathbf{a}_{\rho} + \left( \frac{\partial H_\rho}{\partial z} - \frac{\partial H_z}{\partial \rho} \right) \mathbf{a}_{\phi} + \left( \frac{1}{\rho} \frac{\partial (\rho H_\phi)}{\partial \rho} - \frac{1}{\rho} \frac{\partial H_\rho}{\partial \phi} \right) \mathbf{a}_{z} $$ Given \(\mathbf{H}=20 \rho^{2} \mathbf{a}_{\phi}\), we can compute the partial derivatives in the above expression to find \(\mathbf{J}\).

Now, we need to compute the total current passing through the circular surface \(\rho \leq 1, 0 < \phi < 2\pi, z=0\) in the \(\mathbf{a}_{z}\) direction which requires the integration: $$ I = \int\int_S \mathbf{J} \cdot d\mathbf{S} $$ Where \(S\) is the given circular surface and \(d\mathbf{S}\) denotes the infinitesimal surface vector.

Finally, we compute the total current once again, this time by a line integral around the circular path \(\rho = 1, 0 < \phi < 2 \pi, z = 0\). Using Ampere's law in integral form: $$ \oint_C \mathbf{H} \cdot d\mathbf{l} = I_{enc} $$ Where \(C\) is the given circular path, \(d\mathbf{l}\) denotes the infinitesimal arc length vector, and \(I_{enc}\) is the total enclosed current. By evaluating Steps 1, 2 and 3, we can find the current density \(\mathbf{J}\), the total current passing through the given surface, and verify the total current using line integral.