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4) Electric field for a varying sphere Taken by expanding an old exam question. A conducting sphere of radius a has a net charge -75Q on it. This sphere is surrounded by a concentric, positively charged, nonconducting shell of inner radius 3a and outer radius 4a. That shell has a charge density that's radially symmetric but За 4a -750 Qr varies with radius r. It is given by p(r)=- Ta* 4 There are no charges in between the sphere and the shell. For this configuration: a) What's the net charge on just the shell, Qshel? b) What's magnitude and direction of the E field for radial distances from the center listed as below: i) r;=a/2 ii) iii) r=2a ri=5a

4) Electric field for a varying sphere Taken by expanding an old exam question. A conducting sphere of radius a has a net charge -75Q on it. This sphere is surrounded by a concentric, positively charged, nonconducting shell of inner radius 3a and outer radius 4a. That shell has a charge density that's radially symmetric but За 4a -750 Qr varies with radius r. It is given by p(r)=- Ta* 4 There are no charges in between the sphere and the shell. For this configuration: a) What's the net charge on just the shell, Qshel? b) What's magnitude and direction of the E field for radial distances from the center listed as below: i) r;=a/2 ii) iii) r=2a ri=5a

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### 4) Electric Field for a Varying Sphere #### Problem Description: This problem is inspired by an old exam question. Consider a conducting sphere of radius \(\alpha\) with a net charge of \(-75Q\) on it. Surrounding this sphere is a concentric, positively charged, nonconducting shell with an inner radius \(3\alpha\) and an outer radius \(4\alpha\). The shell possesses a radially symmetric charge density that varies with the radius \(r\) according to the following equation: \[ \rho(r) = \frac{Qr}{\pi \alpha^4} \] There are no charges present in the region between the sphere and the shell. #### Diagram Explanation: The diagram on the right illustrates: - A small conducting sphere at the center with radius \(\alpha\). - The conducting sphere has a charge of \(-75Q\). - Surrounding this is a nonconducting spherical shell with: - Inner radius \(3\alpha\). - Outer radius \(4\alpha\). - The charge density of the shell varies radially as \(\rho(r) = \frac{Qr}{\pi \alpha^4}\). #### Questions: **For this configuration:** a) Determine the net charge on the shell, \(Q_{\text{shell}}\). b) Calculate the magnitude and direction of the electric field (\(E\)) at the following radial distances from the center (\(r\)): - \(r_i = \frac{\alpha}{2}\) - \(r_{ii} = 2\alpha\) - \(r_{iii} = 5\alpha\) c) Using the results from part b, plot the value of the radial component of the electric field for \(r\) ranging from \(0\) to \(5\alpha\). Ensure to clearly define the sign convention for your answers and label critical values. Your plot should be consistent with your answers to part b or adjust accordingly if necessary. --- This educational content is aimed at aiding students in understanding complex electric field configurations involving conducting and nonconducting materials with varying charge densities.
Transcribed Image Text:### 4) Electric Field for a Varying Sphere #### Problem Description: This problem is inspired by an old exam question. Consider a conducting sphere of radius \(\alpha\) with a net charge of \(-75Q\) on it. Surrounding this sphere is a concentric, positively charged, nonconducting shell with an inner radius \(3\alpha\) and an outer radius \(4\alpha\). The shell possesses a radially symmetric charge density that varies with the radius \(r\) according to the following equation: \[ \rho(r) = \frac{Qr}{\pi \alpha^4} \] There are no charges present in the region between the sphere and the shell. #### Diagram Explanation: The diagram on the right illustrates: - A small conducting sphere at the center with radius \(\alpha\). - The conducting sphere has a charge of \(-75Q\). - Surrounding this is a nonconducting spherical shell with: - Inner radius \(3\alpha\). - Outer radius \(4\alpha\). - The charge density of the shell varies radially as \(\rho(r) = \frac{Qr}{\pi \alpha^4}\). #### Questions: **For this configuration:** a) Determine the net charge on the shell, \(Q_{\text{shell}}\). b) Calculate the magnitude and direction of the electric field (\(E\)) at the following radial distances from the center (\(r\)): - \(r_i = \frac{\alpha}{2}\) - \(r_{ii} = 2\alpha\) - \(r_{iii} = 5\alpha\) c) Using the results from part b, plot the value of the radial component of the electric field for \(r\) ranging from \(0\) to \(5\alpha\). Ensure to clearly define the sign convention for your answers and label critical values. Your plot should be consistent with your answers to part b or adjust accordingly if necessary. --- This educational content is aimed at aiding students in understanding complex electric field configurations involving conducting and nonconducting materials with varying charge densities.
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