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प्रश्न
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.
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उत्तर
Mutual Inductance
The ability of production of induced emf in one coil, due to varying current in the neighbouring coil is called mutual inductance.
Magnetic flux, Φ = MI Where, M is called coefficient of mutual induction
Mutual Inductance of Two Long Solenoids
Consider two long solenoids S1 and S2 of same length l, such that solenoid S2 surrounds solenoid S1 completely.
Φ21 = M21I1
Where, M21 is the coefficient of mutual induction of the two solenoids
Magnetic field produced inside solenoid S1 on passing current through it,
B1 = μ0n1I1
Magnetic flux linked with each turn of solenoid S2 will be equal to B1 times the area of cross-section of solenoid S1.
Magnetic flux linked with each turn of the solenoid S2 = B1A
Therefore, total magnetic flux linked with the solenoid S2,
Φ21 = B1A × n2l = μ0n1I1× A× n2l
Φ21 = μ0n1n2lAI1
∴ M21 = μ0n1n2Al
Similarly, the mutual inductance between the two solenoids, when current is passed through solenoid S2 and induced emf is produced in solenoid S1, is given by
M12 = μ0n1n2Al
∴M12 = M21 = M (say)
Hence, coefficient of mutual induction between the two long solenoids
`M = mu_0n_1n_2Al`
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.
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
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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.
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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.
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 ______.
