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Chapter 9: Problem 30

Use Stokes' Theorem to derive the integral form of Faraday's law, $$ \int_{C} \mathbf{E} \cdot d \mathbf{s}=-\frac{\partial}{\partial t} \iint_{D} \mathbf{B} \cdot d \mathbf{S} $$ from the differential form of Maxwell's equations.

Short Answer

Expert verified
By applying Stokes' theorem to the differential form of Faraday's law drawn from Maxwell's equations, we can successfully transform it and arrive at its integral form which is \( \int_{C} \mathbf{E} \cdot d \mathbf{s}=-\frac{\partial}{\partial t} \iint_{D} \mathbf{B} \cdot d \mathbf{S} \)

Step by step solution

01

Recall Maxwell's equations and the differential form of Faraday's law

Applying the differential form of Faraday's law from Maxwell's equation we derive that, \( \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} \)
02

Recognizing Stokes' Theorem and its application

The Stokes' theorem states that for any vector field, its curl integrated over some surface is equal to the line integral of the vector field around the boundary of the surface. That is if \( \mathbf{F} \) is a vector field, then \( \int_{C} \mathbf{F} \cdot d \mathbf{s} = \iint_{D} \nabla \times \mathbf{F} \cdot d \mathbf{S} \), where \( D \) is a surface bounded by the closed contour \( C \).
03

Apply Stokes' Theorem

Stokes' theorem allows us to transform the right hand side of the equation. From \( \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} \), we substitute this into Stokes’s theorem, obtaining \( \iint_{D} (\nabla \times \mathbf{E}) \cdot d \mathbf{S} = -\frac{\partial}{\partial t} \iint_{D} \mathbf{B} \cdot d \mathbf{S} \)
04

Derive the integral form of Faraday's law

On the left side of the equation, by definition of the surface integral of a curl of a vector field over a surface, we can rewrite this as a line integral along the boundary of that surface. Therefore, we obtain the final form of the Faraday's law in integral form: \( \int_{C} \mathbf{E} \cdot d \mathbf{s}=-\frac{\partial}{\partial t} \iint_{D} \mathbf{B} \cdot d \mathbf{S} \)

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Most popular questions from this chapter

Let \(S\) be a closed surface and \(V\) the enclosed volume. Prove Green's first and second identities, respectively. a. \(\int_{S} \phi \nabla \psi \cdot \mathbf{n} d S=\int_{V}\left(\phi \nabla^{2} \psi+\nabla \phi \cdot \nabla \psi\right) d V\). b. \(\int_{S}[\phi \nabla \psi-\psi \nabla \phi] \cdot \mathbf{n} d S=\int_{V}\left(\phi \nabla^{2} \psi-\psi \nabla^{2} \phi\right) d V\).

Use Stokes' Theorem to evaluate the integral $$ \int_{C}-y^{3} d x+x^{3} d y-z^{3} d z $$ for \(C\) the (positively oriented) curve of intersection between the cylinder \(x^{2}+y^{2}=1\) and the plane \(x+y+z=1\)

For cylindrical coordinates, $$ \begin{aligned} x &=r \cos \theta \\ y &=r \sin \theta \\ z &=z \end{aligned} $$ find the scale factors and derive the following expressions: $$ \begin{gathered} \nabla f=\frac{\partial f}{\partial r} \hat{\mathbf{e}}_{r}+\frac{1}{r} \frac{\partial f}{\partial \theta} \hat{\mathbf{e}}_{\theta}+\frac{\partial f}{\partial z} \hat{\mathbf{e}}_{z} \\ \nabla \cdot \mathbf{F}=\frac{1}{r} \frac{\partial\left(r F_{r}\right)}{\partial r}+\frac{1}{r} \frac{\partial F_{\theta}}{\partial \theta}+\frac{\partial F_{z}}{\partial z} \\ \nabla \times \mathbf{F}=\left(\frac{1}{r} \frac{\partial F_{z}}{\partial \theta}-\frac{\partial F_{\theta}}{\partial z}\right) \hat{\mathbf{e}}_{r}+\left(\frac{\partial F_{r}}{\partial z}-\frac{\partial F_{z}}{\partial r}\right) \hat{\mathbf{e}}_{\theta}+\frac{1}{r}\left(\frac{\partial\left(r F_{\theta}\right)}{\partial r}-\frac{\partial F_{r}}{\partial \theta}\right) \\\ \nabla^{2} f=\frac{1}{r} \frac{\partial}{\partial r}\left(r \frac{\partial f}{\partial r}\right)+\frac{1}{r^{2}} \frac{\partial^{2} f}{\partial \theta^{2}}+\frac{\partial^{2} f}{\partial z^{2}}. \end{gathered} $$

For \(\mathbf{r}=x \mathbf{i}+y \mathbf{j}+z \mathbf{k}\) and \(r=|\mathbf{r}|\), simplify the following. a. \(\nabla \times(\mathbf{k} \times \mathbf{r})\). b. \(\nabla \cdot\left(\frac{\mathrm{r}}{r}\right)\). c. \(\nabla \times\left(\frac{\mathbf{r}}{r}\right)\). d. \(\nabla \cdot\left(\frac{\mathrm{r}}{r^{3}}\right)\). e. \(\nabla \times\left(\frac{\mathrm{r}}{r^{3}}\right)\).

Consider the integral \(\int_{C} y^{2} d x-2 x^{2} d y\). Evaluate this integral for the following curves: a. \(C\) is a straight line from \((0,2)\) to \((1,1)\). b. \(C\) is the parabolic curve \(y=x^{2}\) from \((0,0)\) to \((2,4)\). c. \(C\) is the circular path from \((1,0)\) to \((0,1)\) in a clockwise direction.
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