Question

A single circular current loop 10.0 cm in diameter carries a 2.00A current. Suppose that we now connect 1000 of these loops in series within a 500cm length to make a solenoid 500 cm long. What is the magnetic field at the center of this solenoid?

Short answer

Magnetic field at the center of the solenoid is$5.03\times {10}^{-4}T$ .

Step-by-step solution

Step 1:  Definition of Solenoid  

A solenoid is a long coil containing large number of close turns of insulated metal wire.

The magnetic field at the center of the solenoid.

The magnetic field at the center of the solenoid is given by

$\begin{array}{r}B={\mu }_{0}nl\\ B={\mu }_0\frac{N}{L}l\end{array}$

Where, B is the magnetic field at the center of solenoid ,${\mu }_0$ is the permeability of vaccum which is equal to$4\mathrm{\pi }\times {10}^{-7}\mathrm{T}.\mathrm{m}/\mathrm{A}$ , l is the current passing through the solenoid , n is the number of turns per unit length.

Calculation of magnetic field at center of the solenoid 

Given: The current passing through the coil is l = 2.00A .

The length of the solenoid is L = 500cm .

The total number of turns is N = 1000 .

Using

$\begin{array}{r}B={\mu }_{0}nI\\ B={\mu }_0\frac{N}{L}I\end{array}$

Now, putting the values of constants in above expression

$\begin{array}{r}B=\left(4\pi \times {10}^{-7}\mathrm{T}.\mathrm{m}/\mathrm{A}\right)\times \frac{1000}{5\mathrm{m}}\times 2.00\mathrm{A}\\ B=5.03\times {10}^{-4}\mathrm{T}\end{array}$

Thus, the magnetic field at the center of the solenoid is$5.03\times {10}^{-4}T$