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Question: A solenoid has a cross sectional area of \( 6.0\times {{10}^{-4}}{{m}^{2}} \) , consists of 400 turn...

A solenoid has a cross sectional area of 6.0×10−4m26.0\times {{10}^{-4}}{{m}^{2}} , consists of 400 turns per meter, and carries a current 0.4A0.4A are connected to a circumference of the solenoid. The ends of the coil are connected to a 1.5Ω1.5\Omega resistor. Suddenly, a switch gets opened, and the current in the solenoid dies to zero in a time 0.050s0.050s . Find the average current which is passing through the coil during this time.

Explanation

Solution

Solenoid is the term used to define a coil of wire which is used as an electromagnet. It may also refer to any instrument that converts electrical energy to mechanical energy using a solenoid. The device generates a magnetic field from electric current and helps the magnetic field to produce linear motion.

Complete step-by-step answer:
First of all let us discuss the solenoid. A solenoid can be used for slowing down the flow of electricity in a circuit, making themselves an example for an inductor, or an impedance equipment. Cars also have a kind of solenoid that is actually just a motor that happens to work with the use of a coil of wire. The duty of the solenoid is to turn electrical energy into mechanical energy.
So here we can apply the equation,
V=−NdϕdtV=-N\dfrac{d\phi }{dt}
As we all know,
ϕ=B⋅A\phi =B\cdot A
Substituting this in the equation,
=−Nd(BAcos⁡(0))dt=−NAdBdt=-N\dfrac{d\left( BA\cos \left( 0 \right) \right)}{dt}=-NA\dfrac{dB}{dt}
The current flowing will be,
I=−NdBdtAR=−NAμ0nΔIdtRI=-\dfrac{N\dfrac{dB}{dt}A}{R}=-\dfrac{NA\dfrac{{{\mu }_{0}}n\Delta I}{dt}}{R}
Now let us substitute the quantities given in the question into this equation,
I=−10×(6×10−4)(4π×10−7)×400×0.40.0501.5I=-\dfrac{10\times \left( 6\times {{10}^{-4}} \right)\dfrac{\left( 4\pi \times {{10}^{-7}} \right)\times 400\times 0.4}{0.050}}{1.5}
Therefore the final answer is after simplifying this equation,
I=1.6×10−5AI=1.6\times {{10}^{-5}}A
Hence the answer for the question has been obtained.

Note: Solenoid is included in a length of insulated wire which is coiled into a cylinder shape. Current in solenoid creates a stronger magnetic field inside the solenoid than outside. The field lines in this region are parallel and closely spaced which shows the field is highly uniform in its strength and direction.