4. Given a vector function E⃗ = 2r cosϕr̂ - r sinϕϕ̂ + z ẑ in cylindrical coordinates. We would

Step 1: Use Gauss's Law (one of Maxwell's equations) to find the volume charge density producing

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4. Given a vector function \(\vec{E} = 2r \cos\phi \hat{r} - r \sin\phi \hat{\phi} + z \hat{z}\) in cylindrical coordinates. We would like to verify the divergence theorem explicitly using the first octant of a cylinder of radius \(r_0\) and height \(2r_0\) as the volume of integration (see figure). Use the following steps for the purpose: a. Use one of the Maxwell's equations to find the volume charge density producing the given electric field. b. Find the total charge enclosed within the portion of the cylinder specified for this problem. \(\oint \vec{E} \cdot d\vec{A} = \frac{Q_{enclosed}}{\epsilon_0}\) c. Evaluate the flux of the electric field over the entire surface enclosing the portion of the cylinder. (Use the back page if you need extra space!) d. Does your calculation verify the divergence theorem? If not explain what the problem causing the failure of the verification!

          4. Given a vector function \(\vec{E} = 2r \cos\phi \hat{r} - r \sin\phi \hat{\phi} + z \hat{z}\) in cylindrical coordinates. We would like to verify the divergence theorem explicitly using the first octant of a cylinder of radius \(r_0\) and height \(2r_0\) as the volume of integration (see figure). Use the following steps for the purpose:
a. Use one of the Maxwell's equations to find the volume charge density producing the given electric field.
b. Find the total charge enclosed within the portion of the cylinder specified for this problem.
\(\oint \vec{E} \cdot d\vec{A} = \frac{Q_{enclosed}}{\epsilon_0}\)
c. Evaluate the flux of the electric field over the entire surface enclosing the portion of the cylinder. (Use the back page if you need extra space!)
d. Does your calculation verify the divergence theorem? If not explain what the problem causing the failure of the verification!

4. Given a vector function E⃗ = 2r cosϕr̂ - r sinϕϕ̂ + z ẑ in cylindrical coordinates. We would like to verify the divergence theorem explicitly using the first octant of a cylinder of radius r0 and height 2r0 as the volume of integration (see figure). Use the following steps for the purpose:
a. Use one of the Maxwell's equations to find the volume charge density producing the given electric field.
b. Find the total charge enclosed within the portion of the cylinder specified for this problem.
∮E⃗· dA⃗ = (Qenclosed)/(ϵ0)
c. Evaluate the flux of the electric field over the entire surface enclosing the portion of the cylinder. (Use the back page if you need extra space!)
d. Does your calculation verify the divergence theorem? If not explain what the problem causing the failure of the verification!

Added by Duane C.

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