Chapter 2: Q19P (page 78)

Calculate $\nabla \times E$ directly from Eq. 2.8, by the method of Sect. 2.2.2. Refer to Prob. 1.63 if you get stuck.

Short Answer

Expert verified

The required value is $\nabla \times E (r) = 0$

Step by step solution

Determine the expression for the electric field in the two sphere.

Write the formula for the electric field of the volume charge.

$E (r) = \frac{1}{4 {πε}_{0}} \int \frac{ρ (r^{'})}{r^{2}} \bar{r} d_{T'}$

Write the curl of the above function as,

$\nabla \times E (r) = \frac{1}{4 {πε}_{0}} \int ∆ \times \frac{(\bar{r})}{r^{2}} r' {ρd}_{T'}$ ….. (1)

Here, $ε_{0}$ is the permittivity of the free space and $ρ$ is the charge density.

Determine the value of ∆×(r¯)r2.

Solve for $∆ \times \frac{(\bar{r})}{r^{2}}$ as,

localid="1655897481663" $∆ \times \frac{(\bar{r})}{r^{2}} = (\frac{1}{r^{2}} \frac{\partial}{\partial r} \bar{r} + \frac{1}{r \sin θ} \frac{\partial}{\partial ϕ} \bar{ϕ}) \times (r^{n} ϕ) = |\begin{matrix} \bar{r} & \hat{θ} & \hat{ϕ} \\ \frac{1}{r^{2}} \frac{\partial}{\partial r} & \frac{1}{r \sin θ} \frac{\partial}{\partial θ} & \frac{1}{r \sin θ} \frac{\partial}{\partial ϕ} \\ r^{2} & 0 & 0 \end{matrix}| = \hat{r} (0 - 0) - \hat{θ} (0 - \frac{1}{r \sin θ} \frac{\partial}{\partial ϕ} r^{2}) + \hat{ϕ} (0 - \frac{1}{r \sin θ} \frac{\partial}{\partial ϕ} r^{2}) = 0$

Substitute 0 for $\nabla \times \frac{(\hat{r})}{r^{2}}$ in the equation $\nabla \times E (r) = \frac{1}{4 p ε_{0}} \int \nabla \times \frac{(\hat{r})}{r^{2}} r^{'} p d τ$ .

$\nabla \times E (r) = \frac{1}{4 p ε_{0}} (0) = 0$

Therefore, the required value is $\nabla \times E (r) = 0 .$

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Calculate $\nabla \times E$ directly from Eq. 2.8, by the method of Sect. 2.2.2. Refer to Prob. 1.63 if you get stuck.

Short Answer

Step by step solution

Determine the expression for the electric field in the two sphere.

Determine the value of ∆×(r¯)r2.

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Calculate∇×Edirectly from Eq. 2.8, by the method of Sect. 2.2.2. Refer to Prob. 1.63 if you get stuck.

Short Answer

Step by step solution

Determine the expression for the electric field in the two sphere.

Determine the value of ∆×(r¯)r2.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Translational Dynamics

Solid State Physics

Energy Physics

Further Mechanics and Thermal Physics

Linear Momentum

Famous Physicists

Study anywhere. Anytime. Across all devices.

Calculate $\nabla \times E$ directly from Eq. 2.8, by the method of Sect. 2.2.2. Refer to Prob. 1.63 if you get stuck.