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A solenoid has length $0.4\;{\text{cm}}$, radius $1\;{\text{cm}}$ and 400 turns of wire. If a current of $5\;{\text{A}}$ is passed through this solenoid, what is the magnetic field inside the solenoid?
(A) $6.28 \times {10^{ - 1}}\;{\text{T}}$
(B) $6.28 \times {10^{ - 3}}\;{\text{T}}$
(C) $6.28 \times {10^{ - 2}}\;{\text{T}}$
(D) $6.28 \times {10^{ - 6}}\;{\text{T}}$

Last updated date: 22nd Feb 2024
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IVSAT 2024
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Hint: A solenoid is a device consisting of a wire coil, a housing and a shiftable plunger (armature). A magnetic field forms around the coil when an electrical current is introduced, drawing the plunger in. More simply, a solenoid converts mechanical work into electrical energy.
Formula Used:
We will use the following formula to find out the solution to this question
\[B = {\mu _ \circ }ni\]
\[B\] is the magnetic field
\[n\] is the number of turns per unit length
\[i\] is the electric current

Complete Step-by-Step Solution:
The following information is provided to us in the question
The length of the wire, \[l = 0.4 m\]
The radius of the wire, \[r = 1 cm\]
The number of turns in the wire, \[N = 400\]
The current passing through the wire, \[i = 5 A\]
Now, let us find the number of turns per unit length of the wire,
\[n = \dfrac{N}{l} = \dfrac{{400}}{{0.4}} = 1000\]
Now, let us substitute everything in the formula above
\[B = {\mu _ \circ }ni\]
We have
\[B = 4\pi \times {10^{ - 7}} \times {10^3} \times 5\]
\[ \Rightarrow B = 2\pi \times {10^{ - 3}} T\]
Upon further solving, we get
\[\therefore B = 6.28 \times {10^{ - 3}} T\]

Hence, the correct option is (B.)

Additional Information: Inside the solenoid, the magnetic field is uniform in nature and is along the solenoid's axis. At any point directly to the solenoid, the field on the outside is very weak and the field lines cannot be seen near the near vicinity. It is important to note that at every position, the field inside it is parallel to its axis.

Note: Students make a mistake in understanding whether it is the number of turns or number of turns per unit length. The question should be thoroughly read in order to understand what the question states.
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