Q59 of 92 Page 249

A tightly-wound, long solenoid has n turns per unit length, a radius r and carries a current i. A particle having charge q and mass m is projected from a point on the axis in a direction perpendicular to the axis. What can be the maximum speed for which the particle does not strike the solenoid?

Given:


Number of turns per unit length = n


Current = i


Radius = r


Charge of particle = q


Mass = m


Formula used:


Magnetic field inside a solenoid(B) = μ0ni,


where


μ0 = magnetic permeability of vacuum = 4π x 10-7 T m A-1,


n = number of turns per unit length,


i = current carried by the wire


Now, when a particle is projected perpendicular to a magnetic field, it describes a circle. Now, for the particle to not strike to solenoid, the required radius is r/2.


Since it is moving in a circular path, centripetal acceleration = mv2/r,


Where


m = mass of particle,


v = velocity,


r = radius


Force due to the magnetic field = qvB,


Where


q = charge of particle,


v = velocity,


B = magnetic field.


Here, r => r/2.


The centripetal and magnetic forces balance each other.


Hence, using the given data:


=> v = qμ0nir/2m (Ans)


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57

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 center of the solenoid due to this circular current. Integrate this expression under proper limits to find the magnetic field at the center of the solenoid.


(b) Verify that if l >> a, the field tends to B = μ0ni and if a >> l, the field tends to


B=. Interpret these results


 58

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 meter in the solenoid.

 60

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 center 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 center?


 61

A capacitor of capacitance 100 μF is connected to a battery of 20 volts for a long time and then disconnected from it. It is now connected across a long solenoid having 4000 turns per meter. It is found that the potential difference across the capacitor drops to 90% of its maximum value in 2.0 seconds. Estimate the average magnetic field produced at the center of the solenoid during this period.