Advertisements
Advertisements
प्रश्न
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
उत्तर
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
संबंधित प्रश्न
Use this law to obtain the expression for the magnetic field inside an air cored toroid of average radius 'r', having 'n' turns per unit length and carrying a steady current I.
Derive an expression for the mutual inductance of two long co-axial solenoids of same length wound one over the other,
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 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.
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 solenoid of radius a and length l has n turns per unit length. It carries an electric current i. Consider a length dx of the solenoid at a distance x from one end. This contains n dx turns and may be approximated as a circular current i n dx. (a) Write the magnetic field at the centre of the solenoid due to this circular current. Integrate this expression under proper limits to find the magnetic field at the centre of the solenoid. (b) verify that if l >> a, the field tends to B = µ0ni and if a >> l, the field tends to `B =(mu_0nil)/(2a)` . Interpret these results.
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.
A tightly-wound, long solenoid is kept with its axis parallel to a large metal sheet carrying a surface current. The surface current through a width dl of the sheet is Kdl and the number of turns per unit length of the solenoid is n. The magnetic field near the centre of the solenoid is found to be zero. (a) Find the current in the solenoid. (b) If the solenoid is rotated to make its axis perpendicular to the metal sheet, what would be the magnitude of the magnetic field near its centre?
A current of 1.0 A is established in a tightly wound solenoid of radius 2 cm having 1000 turns/metre. Find the magnetic energy stored in each metre of the solenoid.
