91影视

The radius of the solenoid is a.

The number of the turns per unit length is n.

The time independent current in the solenoid is l(t).

The magnetic field outside of the solenoid is zero.

The magnetic flux inside the solenoid is given by,

$蠒=B.A$

Here is the magnetic field and A is the area of the loop.

The magnetic field inside the solenoid is given by,

$\mathbf{B}\mathbf{\left(}\mathbf{s}\mathbf{\right)}\mathbf{=}{\mathbf{渭}}_{\mathbf{0}}\mathbf{nl}$

Here I is the current in the solenoid and is the magnetic permeability.

Consider the loop below,

Here s is the distance.

The change in the magnetic flux is given by,

$$\begin{gathered} -\frac{d蠒}{dt}=鈭�E.dl \\ 鈥�\frac{d蠒}{dt}=E鈭�dl............................\left(1\right) \end{gathered}$$

Here is the small unit of the length of the loop and is the electric field.

Substitute for into equation (1).

$E鈭�dl=-\frac{d}{dt}(B.A)...........\left(2\right)$

When the distance s is smaller than the radius of the solenoid. Then the area of the loop is given by,

$A={\mathrm{蟺蝉}}^2$

Substitute ${\mathrm{蟺蝉}}^2$for A , $2\mathrm{蟺蝉}$for$dl$and${渭}_{0}nl$for B into equation (2).

role="math" localid="1657524125058" $$\begin{gathered} \mathit{E}\mathbf{\left(}\mathbf{2}\mathit{蟺}\mathit{s}\mathbf{\right)}\mathbf{=}\mathbf{-}\frac{\mathbf{d}}{\mathbf{d}\mathbf{t}}\mathbf{(}{\mathbf{渭}}_{\mathbf{0}}\mathit{n}\mathit{l}\mathbf{.}\mathit{蟺}{\mathit{s}}^{\mathbf{2}}\mathbf{)} \\ \mathit{E}\mathbf{\left(}\mathbf{2}\mathit{蟺}\mathit{s}\mathbf{\right)}\mathbf{=}\mathbf{-}{\mathbf{渭}}_{\mathbf{0}}{\mathbf{nTTs}}^{\mathbf{2}}\frac{\mathbf{dl}}{\mathbf{dt}} \\ \mathbf{E}\mathbf{\left(}\mathbf{2}\mathbf{\right)}\mathbf{=}\mathbf{-}{\mathbf{渭}}_{\mathbf{0}}\mathbf{ns}\frac{\mathbf{dl}}{\mathbf{dt}} \\ \mathbf{E}\mathbf{=}\frac{\mathbf{-}{\mathbf{渭}}_{\mathbf{0}}\mathbf{ns}}{\mathbf{2}}\frac{\mathbf{dl}}{\mathbf{dt}} \end{gathered}$$

Hence the electric field at distance inside the loop is $\frac{-{渭}_{0}ns}{2}\frac{dI}{dt}$and direction electric field is same as the current i.e. $\widehat{蠒}$.

When the distance is greater than the radius of the solenoid. Then the area of the loop is given by,

$A={\mathrm{蟺补}}^2$

Substitute ${\mathrm{蟺补}}^2$for A , for $dl$and ${渭}_{0}nl$for B into equation (2).