Advertisements
Advertisements
Question
Derive the expression for the magnetic field due to a solenoid of length ‘2l’, radius ‘a’ having ’n’ number of turns per unit length and carrying a steady current ‘I’ at a point
on the axial line, distance ‘r’ from the centre of the solenoid. How does this expression compare with the axial magnetic field due to a bar magnet of magnetic moment ‘m’?
Advertisements
Solution
Consider a solenoid of length 2l, radius a and having n turns per unit length. It is carrying a current I.
We have to evaluate the axial field at a point P at a distance r from the centre of the solenoid
Consider a circular element of thickness dx of the solenoid at a distance x from its centre.
The magnitude of the field due to this circular loop carrying a current I is given as
`dB=(mu_0dxnIa^2)/(2[(r-x)^2+a^2]^(3/2))`
The magnitude of the total field is obtained by integrating over all the elements from x = −l to x = +l.
`B=(mu_0nIa^2)/2int_(-1)^(+1)dx/(2[(r-x)^2+a^2]^(3/2))`
Consider the far axial field of the solenoid, so r ≫a and r≫l. Hence, we have
[(r-x)2+a2]3/2 ≈ r3
So, we get
`B=(mu_0nIa^2)/(2r^3)int_(-1)^(+1)dx`
`B=(mu_0nIa^2)/(2r^3)2l`
The magnitude of the magnetic moment of the solenoid is
m Total number of turns current cross-sectional area
m=(nx2l)xIx(πa2)
Therefore, we get the magnetic field as
`B=(mu_0)/(4pi) (2m)/r^3`
This is the expression for magnetic field due to a solenoid on the axial line at a distance r from the centre.
This magnetic field is also the field due to a bar magnet of magnetic moment m.
APPEARS IN
RELATED QUESTIONS
Two long coaxial insulated solenoids, S1 and S2 of equal lengths are wound one over the other as shown in the figure. A steady current "I" flow thought the inner solenoid S1 to the other end B, which is connected to the outer solenoid S2 through which the same current "I" flows in the opposite direction so as to come out at end A. If n1 and n2 are the number of turns per unit length, find the magnitude and direction of the net magnetic field at a point (i) inside on the axis and (ii) outside the combined system
Define self-inductance of a coil.
Define mutual inductance between two long coaxial solenoids. Find out the expression for the mutual inductance of inner solenoid of length l having the radius r1 and the number of turns n1 per unit length due to the second outer solenoid of same length and r2 number of turns per unit length.
How is the magnetic field inside a given solenoid made strong?
A long solenoid of radius 2 cm has 100 turns/cm and carries a current of 5 A. A coil of radius 1 cm having 100 turns and a total resistance of 20 Ω is placed inside the solenoid coaxially. The coil is connected to a galvanometer. If the current in the solenoid is reversed in direction, find the charge flown through the galvanometer.
A circular coil of one turn of radius 5.0 cm is rotated about a diameter with a constant angular speed of 80 revolutions per minute. A uniform magnetic field B = 0.010 T exists in a direction perpendicular to the axis of rotation. Suppose the ends of the coil are connected to a resistance of 100 Ω. Neglecting the resistance of the coil, find the heat produced in the circuit in one minute.
The magnetic field B inside a long solenoid, carrying a current of 5.00 A, is 3.14 × 10−2 T. Find the number of turns per unit length of the solenoid.
A long solenoid is fabricated by closely winding a wire of radius 0.5 mm over a cylindrical nonmagnetic frame so that the successive turns nearly touch each other. What would be the magnetic field B at the centre of the solenoid if it carries a current of 5 A?
A tightly-wound, long solenoid carries a current of 2.00 A. An electron is found to execute a uniform circular motion inside the solenoid with a frequency of 1.00 × 108 rev s−1. Find the number of turns per metre in the solenoid.
The length of a solenoid is 0.4 m and the number turns in it is 500. A current of 3 amp, is flowing in it. In a small coil of radius 0.01 m and number of turns 10, a current of 0.4 amp. is flowing. The torque necessary to keep the axis of this coil perpendicular to the axis of solenoid will be ______.
