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प्रश्न
A metallic rod of length ‘l’ is rotated with a frequency v with one end hinged at the centre and the other end at the circumference of a circular metallic ring of radius r, about an axis passing through the centre and perpendicular to the plane of the ring. A constant uniform magnetic field B parallel to the axis is present everywhere. Using Lorentz force, explain how emf is induced between the centre and the metallic ring and hence obtained the expression for it.
उत्तर
Suppose the length of the rod is greater than the radius of the circle and rod rotates anticlockwise and suppose the direction of electrons in the rod at any instant be along +y-direction.
Suppose the direction of the magnetic field is along +z-direction.
Then, using Lorentz law, we get the following:
`vecF = -e (vecv xx vecB)`
`=> vecF = -e(vhatj xx Bhatk)`
`=> hatF = -evBhati`
Thus, the direction of force on the electrons is along −x axis.
Thus, the electrons will move towards the center i.e., the fixed end of the rod. This movement of electrons will result in current and hence it will produce emf in the rod between the fixed end and the point touching the ring.
Let θ be the angle between the rod and radius of the circle at any time t.
Then, area swept by the rod inside the circle `= 1/2 πr^2θ`
Inducced emf ` = B xx d/dt (1/2πr^2θ) =1/2πr^2B (dθ)/dt = 1/2πr^2Bω = 1/2πr^2B (2πv) = π^2r^2Bv`
NOTE: There will be an induced emf between the two ends of the rods also.
संबंधित प्रश्न
Consider the following statements:-
(A) An emf can be induced by moving a conductor in a magnetic field.
(B) An emf can be induced by changing the magnetic field.
Consider the situation shown in figure. The wire AB is slid on the fixed rails with a constant velocity. If the wire AB is replaced by a semicircular wire, the magnitude of the induced current will _____________ .
The flux of magnetic field through a closed conducting loop changes with time according to the equation, Φ = at2 + bt + c. (a) Write the SI units of a, b and c. (b) If the magnitudes of a, b and c are 0.20, 0.40 and 0.60 respectively, find the induced emf at t = 2 s.
A conducting loop of area 5.0 cm2 is placed in a magnetic field which varies sinusoidally with time as B = B0 sin ωt where B0 = 0.20 T and ω = 300 s−1. The normal to the coil makes an angle of 60° with the field. Find (a) the maximum emf induced in the coil, (b) the emf induced at τ = (π/900)s and (c) the emf induced at t = (π/600) s.
The north pole of a magnet is brought down along the axis of a horizontal circular coil (see the following figure). As a result, the flux through the coil changes from 0.35 weber to 0.85 weber in an interval of half a second. Find the average emf induced during this period. Is the induced current clockwise or anticlockwise as you look into the coil from the side of the magnet ?
A wire-loop confined in a plane is rotated in its own plane with some angular velocity. A uniform magnetic field exists in the region. Find the emf induced in the loop.
A 10 m wide spacecraft moves through the interstellar space at a speed 3 × 107 m s−1. A magnetic field B = 3 × 10−10 T exists in the space in a direction perpendicular to the plane of motion. Treating the spacecraft as a conductor, calculate the emf induced across its width.
The two rails of a railway track, insulated from each other and from the ground, are connected to a millivoltmeter. What will be the reading of the millivoltmeter when a train travels on the track at a speed of 180 km h−1? The vertical component of earth's magnetic field is 0.2 × 10−4 T and the rails are separated by 1 m.
The current in a solenoid of 240 turns, having a length of 12 cm and a radius of 2 cm, changes at a rate of 0.8 A s−1. Find the emf induced in it.
A rectangular loop of sides 8 cm and 2 cm with a small cut is stationary in a uniform magnetic field directed normal to the loop. The magnetic field is reduced from its initial value of 0.3 T at the rate of 0.02 T s-1 If the cut is joined and loop has a resistance of 1.6 Ω, then how much power is dissipated by the loop as heat?
