Q7P - Introduction to Electrodynamics

Question

Find the electric field a distance z from the center of a spherical surface of radius R (Fig. 2.11) that carries a uniform charge density $\mathit{\sigma }$. Treat the case z < R (inside) as well as z > R (outside). Express your answers in terms of the total charge q on the sphere. [Hint: Use the law of cosines to write $\mathit{r}$ in terms of R and $\mathit{\theta }$. Be sure to take the positive square root: $\sqrt{{\mathbf{R}}^{\mathbf{2}}\mathbf{+}{\mathbf{z}}^{\mathbf{2}}\mathbf{-}\mathbf{2}\mathbf{R}\mathbf{z}}\mathbf{=}\left(R-z\right)$ if $\mathit{R}\mathbf{>}\mathit{z}$, but it's $\left(z-R\right)\mathbf{ }$if $\mathit{R}\mathbf{<}\mathit{z}$.]

Step-by-Step Solution

Answer

The electric field outside the sphere$\left(z>R\right)$ is $E=\frac{1}{4\pi {\epsilon }_0} \frac{q}{z^2}$. The electric field outside the sphere$\left(z>R\right)$ is $E=0$.

Step 1: Write the given information.

The radius of the spherical surface is $R$.

The uniform surface charge density $\sigma$.

Step 2: Define the coulomb&rsquo;s law.

Electric field due to charge   at a distance   is proportional to the charge  and inversely proportional to the square of the distance  as,

$\mathit{E}\mathbf{=}\frac{\mathbf{1}}{\mathbf{4}\mathbf{\pi }{\mathbf{\epsilon }}_{\mathbf{0}}}\mathbf{ }\frac{\mathbf{q}}{{\mathbf{z}}^{\mathbf{2}}}$

Step 3: Draw the Gaussian surface.

The uniformly charged spherical surface of radius$R$, which carries a surface density $\sigma$is drawn. At a distance r from z axis, a infinitesimal area is drawn, as shown below:

                     

Here $\theta ,\varphi ,\phi$are the angles with respect to $z$,$x$ and $y$ axis respectively. From the above, diagram using Pythagoras theorem in right triangle.

$$\begin{gathered} r^2={\left(z-R \cos \varphi \right)}^2+{\left(R \sin \varphi \right)}^2 \\ =z^2+R^2{\cos }^2\varphi -2Rz\cos \varphi +R^2 {\sin }^2\theta \\ =z^2+R^2-2Rz\cos \varphi \\ r=\sqrt{z^2+R^2-2Rz\cos \varphi } \end{gathered}$$

Solve further as,

$\cos \theta =\frac{z-R \cos \theta }{r}$

Step 4: Obtain the expression for electric field

The differential surface charge is obtained by multiplying density with the differential area as

$dq=\sigma R^2\sin \theta d\theta d\varphi$

The differential field due to differential charge on area   can be written as

$dE=\frac{1}{4\pi {\epsilon }_0} \frac{dq}{r^2}\cos \psi$.

$$\begin{gathered} Substitute \sqrt{z^2+R^2-2Rz\cos \varphi } for r,\frac{2Rz\cos \theta }{r}for \cos \psi and \\ \sigma R^2\sin \theta d\theta d\varphi for dqinto the equation. \end{gathered}$$

$$\begin{gathered} dE=\frac{1}{4\pi {\epsilon }_0} \frac{\left(\sigma R^2\sin \theta d\theta d\varphi \right)}{{\left(\sqrt{z^2+R-2Rz\cos \varphi } \right)}^2}\left(\frac{z-R\cos \theta }{\sqrt{z^2-2Rz\cos \varphi }}\right) \\ =\frac{1}{4\pi {\epsilon }_0} \frac{\left(\sigma R^2\sin \theta d\theta d\varphi \right)\left(z-R\cos \theta \right)}{{\left(z^2+R^2-2Rz\cos \varphi \right)}^{3/2}} \end{gathered}$$

Integrate above differential integral as,

$$\begin{gathered} E=\int dE \\ =\frac{1}{4\pi {\epsilon }_0}{\int }_0^{2\pi }d\varphi {\int }_0^{\pi }\frac{\left(\sigma R^2\sin \theta d\theta \right)\left(z-R\cos \theta \right)}{{\left(z^2+R^2-2Rz\cos \theta \right)}^{3/2}} \\ =\frac{1}{4\pi {\epsilon }_0}\left(2\pi \right){\int }_0^{\pi }\frac{\left(\sigma R^2\sin \theta d\theta \right)\left(z-R\cos \theta \right)}{{\left(z^2+R^2-2Rz\cos \theta \right)}^{3/2}} \\ =\frac{2\pi \sigma R^2}{4\pi {\epsilon }_0}{\int }_0^{\pi }\frac{\left(z-R\cos \theta \right) \left(\sin \theta d\theta \right)}{{\left(z^2+R^2-2Rz\cos \theta \right)}^{3/2}} \end{gathered}$$

Consider that $u=\cos \theta$, such that $du=-\sin \theta d\theta$. For $\theta =\pi$, $u=-1$ and  $\theta =0$, $u=1$

Substitute $u$ for $\cos \theta$, $-\sin \theta d\theta$ for $du$ into

$E=\frac{2\pi \sigma R^2}{4\pi {\epsilon }_0}{\int }_0^{\pi }\frac{\left(z-R\cos \theta \right) \left(\sin \theta d\theta \right)}{{\left(z^2+R^2-2Rz\cos \varphi \right)}^{3/2}}$.

$E=\frac{2\pi \sigma R^2}{4\pi {\epsilon }_0}{\int }_1^{-1}\frac{\left(z-Ru\right) \left(du\right)}{{\left(z^2+R^2-2Rzu\right)}^{3/2}}$

Consider that $Ru=y$, such that $du=\frac{dy}{R}$.

Substitute $y$ for $Ru$, $\frac{dy}{R}$ for into 

$E=\frac{2\pi \sigma R^2}{4\pi {\epsilon }_0}{\int }_1^{-1}\frac{\left(z-Ru\right) \left(du\right)}{{\left(z^2+R^2-2Rzu\right)}^{3/2}}$

$E=\frac{2\pi \sigma R}{4\pi {\epsilon }_0}{\int }_1^{-1}\frac{\left(z-y\right) \left(dy\right)}{{\left(z^2+R^2-2zy\right)}^{3/2}}$

Use the formula $\int \frac{\left(z-y\right) \left(dy\right)}{{\left(z^2+R^2-2zy\right)}^{3/2}}=\left(\frac{yz-R^2}{z^2\sqrt{R^2+Z^{2-2y}}}\right)$,to evaluate above integral as$E=\frac{2\pi \sigma R}{4\pi {\epsilon }_0}\left(\frac{yz-R^2}{z^2\sqrt{R^2+z^2-2y}}\right)$

Substitute back $Ru$ for $y$ into $E=\frac{2\pi \sigma R}{4\pi {\epsilon }_0}\left(\frac{yz-R^2}{z^2\sqrt{R^2+z^2-2y}}\right)$ as,$E=\frac{2\pi \sigma R}{4\pi {\epsilon }_0}\left(\frac{Ruz-R^2}{z^2\sqrt{R^2+z^2-2y}}\right)$

Apply the limits from $-1$ to 1 into above equation.

$$\begin{gathered} E=\frac{2\pi \sigma R^2}{4\pi {\epsilon }_0}\left(\frac{uz-R}{z^2\sqrt{R^2+z^2-2yRzu}}\right) \\ =\frac{2\pi \sigma R^2}{4\pi {\epsilon }_0{z}^2}\left[\left(\frac{z-R}{\sqrt{R^2+z^2-2Rz}}\right)-\left(\frac{z-R}{\sqrt{R^2+z^2-2Rz}}\right)\right] \\ =\frac{2\pi \sigma R^2}{4\pi {\epsilon }_0{z}^2}\left[\left(\frac{z-R}{\left|z-R\right|}\right)-\left(\frac{z-R}{\left|z-R\right|}\right)\right] \\ =\frac{2\pi \sigma R^2}{4\pi {\epsilon }_0{z}^2}\left[\left(\frac{z-R}{\left|z-R\right|}\right)+\left(\frac{z-R}{\left|z-R\right|}\right)\right] \end{gathered}$$

Step 5: Obtain the electric field for z &#62; R

For $z>R$ values of $z-R$ and $z+R$ can be approximated to $Z$ only.

Substitute $Z$ for$Z-R$, $Z+R$, and $\frac{q}{4\pi R^2}$ for $\sigma$ into ,

$$\begin{gathered} E=\frac{2\pi \sigma R^2}{4\pi {\epsilon }_0{z}^2}\left[\left(\frac{z-R}{\left|z-R\right|}\right)+\left(\frac{z+R}{\left|z+R\right|}\right)\right] \\ E=\frac{2\pi \sigma R^2}{4\pi {\epsilon }_0{z}^2}\left[\left(\frac{z}{\left|z\right|}\right)+\left(\frac{z}{\left|z\right|}\right)\right] \\ =\frac{4\pi \left({\displaystyle \frac{q}{4\pi R^2}}\right)R^2}{4\pi {\epsilon }_0{z}^2} \\ =\frac{1}{4\pi {\epsilon }_0}\frac{q}{z^2} \end{gathered}$$

Thus, the electric field outside the sphere is $E=\frac{1}{4\pi {\epsilon }_0}\frac{q}{z^2}$.

Step 6: Obtain the electric field for z &#60; R

For $z<R$ values of $z-R$ and $z+R$ can be approximated to $-R$ and $R$ respectively.

Substitute $-R$ for $z-R$, $R$ for $z+R$ into, $E=\frac{2\pi \sigma R^2}{4\pi {\epsilon }_0{z}^2}\left[\left(\frac{z-R}{\left|z-R\right|}\right)+\left(\frac{z+R}{\left|z+R\right|}\right)\right]$.

$$\begin{gathered} E=\frac{2\pi \sigma R^2}{4\pi {\epsilon }_0{z}^2}\left[\left(\frac{-R}{\left|-R\right|}\right)+\left(\frac{zR}{\left|R\right|}\right)\right] \\ =0 \end{gathered}$$

Thus, the electric field outside the sphere is $E=0$.