Question 5. V for Spherical shell at P. Work the integral as done in class and obtain In class we obtained the potential for a thin, uniformly charged a a Spherical Shell by Direct Integration a the results for the Potential inside and outside of the shell. Recall Q dq = 0,dA %3D %3D dq = 0, R? sin 0 de do r = VR2 + z² – 2zR cos 0 %3! R O,R2 sin Ode do V = ke | VR2 + z2 – 2zR cos e | Voutside (z > R) = ke Vinside (z < R) = ke %3D
Question 5. V for Spherical shell at P. Work the integral as done in class and obtain In class we obtained the potential for a thin, uniformly charged a a Spherical Shell by Direct Integration a the results for the Potential inside and outside of the shell. Recall Q dq = 0,dA %3D %3D dq = 0, R? sin 0 de do r = VR2 + z² – 2zR cos 0 %3! R O,R2 sin Ode do V = ke | VR2 + z2 – 2zR cos e | Voutside (z > R) = ke Vinside (z < R) = ke %3D
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![**Question 5: \( V \) for a Spherical Shell by Direct Integration**
In class, we obtained the potential for a thin, uniformly charged spherical shell at \( P \). Work the integrals as done in class and obtain the results for the potential inside and outside of the shell. Recall \( Q = \sigma_0 4 \pi R^2 \).
\[ dq = \sigma_0 R^2 \sin \theta \, d\theta \, d\phi \]
\[ r = \sqrt{R^2 + z^2 - 2zR \cos \theta} \]
\[ V = k_e \iint \frac{\sigma_0 R^2 \sin \theta \, d\theta \, d\phi}{\sqrt{R^2 + z^2 - 2zR \cos \theta}} \]
\[ V_{\text{outside}} (z > R) = k_e \frac{Q}{z} \]
\[ V_{\text{inside}} (z < R) = k_e \frac{Q}{R} \]
**Diagram Explanation:**
The diagram shows a sphere with radius \( R \) and a point \( P \) located outside the sphere at distance \( z \) from the center. The angle \( \theta \) is measured from the center to the point on the surface, and the charge element \( dq = \sigma_0 dA \) is depicted on the shell's surface.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F249fe9e2-5f1e-4e39-9ada-88c965da2dda%2Ff43522e5-1bfe-44af-8164-9ebe5031ade2%2F6vsgntr_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Question 5: \( V \) for a Spherical Shell by Direct Integration**
In class, we obtained the potential for a thin, uniformly charged spherical shell at \( P \). Work the integrals as done in class and obtain the results for the potential inside and outside of the shell. Recall \( Q = \sigma_0 4 \pi R^2 \).
\[ dq = \sigma_0 R^2 \sin \theta \, d\theta \, d\phi \]
\[ r = \sqrt{R^2 + z^2 - 2zR \cos \theta} \]
\[ V = k_e \iint \frac{\sigma_0 R^2 \sin \theta \, d\theta \, d\phi}{\sqrt{R^2 + z^2 - 2zR \cos \theta}} \]
\[ V_{\text{outside}} (z > R) = k_e \frac{Q}{z} \]
\[ V_{\text{inside}} (z < R) = k_e \frac{Q}{R} \]
**Diagram Explanation:**
The diagram shows a sphere with radius \( R \) and a point \( P \) located outside the sphere at distance \( z \) from the center. The angle \( \theta \) is measured from the center to the point on the surface, and the charge element \( dq = \sigma_0 dA \) is depicted on the shell's surface.
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V for spherical shell
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