Question

A solenoid with 385 turns per meter and a diameter of 17.0 cm has a magnetic flux through its core of magnitude 1.28 × 10-4 (T·m2 ). (a) Find the current in this solenoid. (b) How would your answer in part (a) change if the diameter of the solenoid were doubled? Explain your detailed reasoning and give the answer.

Answer 1

Answer:

(a)The current passes through the solenoid is 11.7 A.

(b) The new current will be one-fourth of the initial current.

Explanation:

Given that,

The number of turns per meter = 385

Diameter of solenoid = 17.0 cm = $17\times 10^{-2}$ m

Magnetic flux  through core of solenoid $\phi$ = $1.28\times 10^{-4}$ Tm²

(a)

Magnetic field B= $\mu_0nI$

$\mu_0= 4\pi \times 10^{-7}$ T/amp m

Cross section area of the solenoid A= $\pi \frac{d^2}{4}$

                                                             $=\pi\frac{ (17\times 10^{-2})^2}{4}$  m²

The angle between magnetic field and cross section of the solenoid is $\theta =0^\circ$

The magnetic flux through a area A with magnetic fie;d B is

$\phi = BA cos\theta$

$\Rightarrow \phi =( \mu_0nI)(\pi \frac{d^2}4)cos \theta$

$\Rightarrow I =\frac{\phi}{(\mu_0\pi n \frac{d^2}4cos\theta)}$

       $=\frac{4\phi}{(\mu_0n)(\pi d^2)cos\theta}$

      $=\frac{1.28\times 10^{-4}\times 4}{(4\pi \times10^{-7}\times385 )\times(\pi\times17\times 10^{-2})^2cos 0^\circ}$

     =11.7 A

The current of the solenoid is 11.7 A.

(b)

$I =\frac{4\phi}{(\mu_0\pi n d^2cos\theta)}$

From the above equation it is clear that, the current is inversely proportional to the square of the diameter of a solenoid.

$I\propto \frac1{d^2}$.

Consider d' be the new diameter of the solenoid .

Since the new diameter of the solenoid is double of the initial diameter.

That is d'= 2d.

$\frac{I}{I'}= \frac{(d')^2}{d^2}$

$\Rightarrow \frac{I}{I'}=\frac{(2d)^2}{d^2}$

$\Rightarrow \frac{I}{I'}=4$

⇒I=4I'

$\Rightarrow I'=\frac{I}{4}$

The new current will be one-fourth of the initial current.