91Ó°ÊÓ

The solenoid is the current carrying wire. The electric field circles around inside the solenoid.

So at a distance $r<R$.

$∫E·ds=E\left(2\mathrm{π}r\right)$

And area $A=\mathrm{Ï€}{r}^2$

When magnetic field varies sinusoidally the expression for magnetic field is.

$B\left(t\right)=B_c+B_0\sin \left(2\mathrm{Ï€}ft\right)$

Substitute $10.0\mathrm{T}$for $B_c$, $2.0\mathrm{T}$for $B_0$and $10Hz$for $f$.

role="math" localid="1649862479221" $B\left(t\right)=(10.0T)+(2.0T)\sin \left(2\mathrm{Ï€}\right(10\mathrm{Hz}\left)\mathrm{t}\right)$

role="math" localid="1649862488387" $=(10.0\mathrm{T})+(2.0\mathrm{T})\sin \left(20\mathrm{Ï€}t\right)$

Differentiate $B\left(t\right)$

role="math" localid="1649862500756" $\frac{d\left(B\right(t\left)\right)}{dt}=\left(40\mathrm{Ï€}\mathrm{T}\right)\cos \left(20\mathrm{Ï€}t\right)$

Substitute $\mathrm{π}{r}^2$for $A,E·\left(2\mathrm{π}r\right)$for $∫E·ds$, and role="math" localid="1649862517441" $\left(40\mathrm{π}\mathrm{T}\right)\cos \left(20\mathrm{π}t\right)$for $d\left(B\right(t\left)\right)/dt$

role="math" localid="1649862530672" $E·\left(2\mathrm{π}r\right)=-\left(\right(40\mathrm{π}\mathrm{T}\left)\cos \right(20\mathrm{π}t\left)\right)\left(\mathrm{π}{r}^2\right)$

role="math" localid="1649862541580" $E=-\frac{\left(\right(40\mathrm{Ï€}\mathrm{T}\left)\cos \right(20\mathrm{Ï€}t\left)\right)\left(\mathrm{Ï€}r\right)}{\left(2\mathrm{Ï€}\right)}$

The electric field will be maximum when role="math" localid="1649862607430" $\cos \left(20\mathrm{Ï€³Ù}\right)=1$

Substitute $\cos \left(20\mathrm{Ï€³Ù}\right)=1$(to write the above expression in term of$E_{max}$).

$E_{\max }=\frac{\left(\right(40\mathrm{π}\mathrm{T}\left)1\right)·\left(\mathrm{π}r\right)}{\left(2\mathrm{π}\right)}$

Substitute $1.5cm$for $r$to find $E_{max}$.

$E_{\max }=-\frac{\left(40\mathrm{Ï€}\mathrm{T}\right)\left(1.5\mathrm{cm}\left(\frac{1\mathrm{m}}{{10}^2\mathrm{cm}}\right)\right)}{\left(2\mathrm{Ï€}\right)}$

$=0.3V/m$

The electric field reaches maximum when,

$22\mathrm{Ï€³Ù}=\mathrm{Ï€},3\mathrm{Ï€},5\mathrm{Ï€}$,....

Where sine function goes to zero.

So when electric field reaches maximum,

$B=B_e+B_{o_{}}\sin \left(\mathrm{Ï€}\right)$

$=B_e$

$=10T$