Question

3. A magnetic field which is uniform along the z-axis is given by \vec{B}(x, y)=\frac{1}{1+x^{2}} \mathbf{i}_{\mathbf{x}}+\frac{2 x y}{\left(1+x^{2}\right)^{2}} \mathbf{i}_{y} . )Show that the magnetic field given above satisfies Gauss' Law for magnetism,V.B = 0. ) Consider a particle with charge, q, moving with velocity, 7 = voi,, along the x-axis. Calculate the Lorentz force acting on the charged particle. (c) A rigid conductor carrying a current, I, is placed in the magnetic field. It is aligned parallel with the x-axis, with one end at x =positon a = 1, as shown in Fig. 2. Show that the force on the wire is given by \vec{F}=I y \int_{0}^{l} \frac{2 x}{\left(1+x^{2}\right)^{2}} d x \mathbf{i}_{z} ) Use the substitution u= 1+x to show that this integral can be evaluated to give

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