$FIGUREP31.39$Shows the electric field inside a cylinder of radius localid="1649879543758" $R=3.0mm.$. The field strength is increasing with time as localid="1649879549377" $E=1.0\times {10}^8{t}^{2}V/m$, where $t$is in $s$. The electric field outside the cylinder is always zero, and the field inside the cylinder was zero for localid="1649879554833" $t<0$.

localid="1649879566359" $a.$Find an expression for the electric flux localid="1649879560575" ${\varphi }_e$through the entire cylinder as a function of time.

localid="1649879572467" $b$. Draw a picture showing the magnetic field lines inside and outside the cylinder. Be sure to include arrowheads showing the field’s direction.

localid="1649879587867" $c$. Find an expression for the magnetic field strength as a function of time at a distance localid="1649879577475" $r<R$from the center. Evaluate the magnetic field strength atlocalid="1649879582437" $r=2.0mm$, localid="1649879592577" $t=2.0s$.

d. Find an expression for the magnetic field strength as a function of time at a distancelocalid="1649879597467" $r>R$from the center. Evaluate the magnetic field strength at localid="1649879602321" $r=4.0mm$, localid="1649879607075" $t=2.0s$.

We have given a cylinder of $R=3.0mm$and a time varying electric field as $E=1.0\times {10}^8\times t^2\raisebox{1ex}{$V$}\!\left/ \!\raisebox{-1ex}{$m$}\right..$Where the electric field outside the cylinder is always zero, and the field inside the cylinder was zero for $t<0$. We need to find expression for the electric flux ${\phi }_e$through the entire cylinder as a function of time.

Now in this part(a) we are asked to find out the electric flux (${\phi }_e$) passing through the entire cylinder.

Now as we know . ${\phi }_e=\stackrel{\rightharpoonup }{E}.\stackrel{\rightharpoonup }{A}$

So,

Area = localid="1649881722630" $\mathrm{\pi }{R}^2=\mathrm{\pi }\times {\left(3\right)}^2=9\mathrm{\pi }\times {10}^{-6}{m}^2$

Now,

localid="1649882685816" $$\begin{gathered} \Rightarrow {\varphi }_e=1.0\times {10}^{8_{}}\times t^2\times 9\mathrm{\pi }\times {10}^{-6} \\ \Rightarrow {\varphi }_e=(9\mathrm{\pi }\times {10}^2\times t^2)V/m. \end{gathered}$$

We are given of a cylindrical region and $E=1.0\times {10}^8\times t^{2}V/m.$Now we need to draw a picture showing the magnetic field lines inside and outside the cylinder. Also we need to draw the arrowheads showing the field direction

Here by using right hand thumb rule the magnetic field lines are oriented in clockwise circles both inside and outside the cylinder

We are given a cylindrical region and $E=1.0\times {10}^8\times t^{2}V/m.$We need to find an expression for the magnetic field strength as a function of time at a distance $r<R$from the center. Evaluate the magnetic field strength at $r=2.0mm$, $t=2.0s$

We need to find out the expression for the magnetic field strength as a function of time at a distance$r>R$ from the center. Evaluate the magnetic field strength at $r=4.0mm,t=2.0s$

By using Ampere-Maxwell law, We will consider a loop of radius $r$outside the circle of radius$R$(where$r>R$

localid="1649882721012" $$\begin{gathered} \Rightarrow B\left(r\right).2\mathrm{\pi }r=\frac{1}{c^2}\mathrm{\pi }{R}^2\frac{dE}{dt}=\frac{1}{c^2}\mathrm{\pi }{R}^2(1\times {10}^8)\times 2t \\ \Rightarrow B\left(r\right)=\frac{(1.0\times {10}^8)}{c^2}t\times R^2 \\ \Rightarrow B=5.0\times {10}^{-12}T \end{gathered}$$