Physics

Physics · electromagnetism

Problem

8. Calculate the magnetic field due to the current in question 6. Assume that the permeability of the liquid is equal to that of vacuum μ=μ0\mu=\mu_{0}. Notes ∫αdxα2+x2=arctan⁡xa+\int \frac{\alpha d x}{\alpha^{2}+x^{2}}=\arctan \frac{x}{a}+ const Given constants: A system consisted of two conductor bodies is immersed in a uniform dielectric and weakly conducting liquid. When a constant voltage difference is applied between both conductors, the system has both electric and magnetic fields. In this problem we will investigate this system. Context question: 1. First consider an infinitely long line with charge per unit length λ\lambda in vacuum. Calculate the electric field E(r)\mathbf{E}(\mathbf{r}) due to the line. Context answer: E=r^λ2πϵ0r\mathbf{E}=\hat{r} \frac{\lambda}{2 \pi \epsilon_{0} r} Context question: 2. The potential due to the line charge could be written as V(r)=f(r)+K,V(r)=f(r)+K, where KK is a constant. Determine f(r)f(r). Context answer: f(r)=−λ2πϵ0ln⁡rf(r)=-\frac{\lambda}{2 \pi \epsilon_{0}} \ln r Context question: 3. Calculate the potential in all space V(x,y,z)V(x, y, z) due to an infinitely long line with charge per unit length λ\lambda at x=−b,y=0x=-b, y=0 and another infinitely long line with charge per unit length −λ-\lambda at x=b,y=0x=b, y=0. Both lines are parallel to the zz-axis. Take V=0V=0 at the origin. Sketch the equipotential surfaces. Context answer: <image_1> Extra Supplementary Reading Materials: For the following questions, ignore any edge effects. Context question: 4. Now consider two identical conducting cylinders, both with radius R=3aR=3 a in vacuum. The length of each cylinders are the same and much larger than its radius (l≫R)(l \gg R). The axis of both cylinders are on the xzx z-plane and parallel to the zz-axis, one at x=−5a,y=0x=-5 a, y=0 and the other at x=5a,y=0x=5 a, y=0. An electrical potential difference of V0V_{0} is applied between the two cylinders (the cylinder at x=−5ax=-5 a has the higher potential) by connecting them to a battery. Calculate the potential in all regions. Take V=0V=0 at the origin. Context answer: the potential outside the two cylinders are V=V04ln⁡3ln⁡(4a−x)2+y2(4a+x)2+y2V=\frac{V_{0}}{4 \ln 3} \ln \frac{(4 a-x)^{2}+y^{2}}{(4 a+x)^{2}+y^{2}} The potential inside the cylinder centered at (x=5a,y=0)(x=5 a, y=0) is V=−V0/2V=-V_{0} / 2. The potential inside the cylinder centered at (x=−5a,y=0)(x=-5 a, y=0) is V=V0/2V=V_{0} / 2. Context question: 5. Calculate the capacitance CC of the system. Context answer: lπϵ0ln⁡3\frac{l \pi \epsilon_{0}}{\ln 3} Context question: 6. Now both cylinders are totally immersed in a weakly conducting liquid with conductivity σ\sigma. Calculate the total current that flows between both cylinders. Assume the permittivity of the liquid is equal to that of vacuum, ϵ=ϵ0\epsilon=\epsilon_{0}. Context answer: I=V0πσlln⁡3I=\frac{V_{0} \pi \sigma l}{\ln 3} Context question: 7. Calculate the resistance RR of the system. Calculate RCR C of the system. Context answer: RC=ϵ0σR C=\frac{\epsilon_{0}}{\sigma}

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