Q28 of 91 Page 230

A conducting wire of length ℓ, lying normal to a magnetic field B, moves with velocity v as shown in figure.

(a) Find the average magnetic force on a free electron of the wire.


(b) Due to this magnetic force, electrons concentrate at one end resulting in the electric field inside the wire. The redistribution stops when the electric force on the free electrons balances the magnetic force. Find the electric field developed inside the wire when the redistribution stops.


(c) What potential difference is developed between the ends of the wire?



Given-



Length of the conducting wire = l



Inward magnetic field = B



Velocity of the conducting wire = v



As the wire is moving with velocity v, we can take this as the motion of free electrons present inside the wire with velocity v.



(a) The average magnetic force on a free electron of the wire



We know, Lorentz force F is given by -



where,


e = charge on an electron


v = velocity of the electron


B=magnetic field



(b)The redistribution of electrons stops when the electric force is just balanced by the magnetic force.



Electric force coulomb’s law,



where


e = charge


E=electric field of the charge


and also magnetic force, we know, Lorentz force F is given by -



where,


e = charge on an electron


v = velocity of the electron


B=magnetic field



On equating these two forces, we get-






(c) The potential difference developed between the ends of the wire , V is -




where l is length of wire and E is applied electric field


From (1)



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Suppose that the radius of cross section of the wire used in the previous problem is r. Find the increase in the radius of the loop if the magnetic field is switched off. The Young modulus of the material of the wire is Y.

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The magnetic field existing in a region is given by


A square loop edge ℓ and carrying a current i, is placed with its edges parallel to the x-y axes. Find the magnitude of the net magnetic force experienced by the loop.


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A current i is passed through a silver strip of width d and area of cross section A. The number of free electrons per unit volume is π.

(a) Find the drift velocity v of the electrons.


(b) If a magnetic field B exists in the region as shown in figure, what is the average magnetic force on the free electrons?


(c) Due to the magnetic force, the free electrons get accumulated on one side of the conductor along its length. This produces a transverse electric field in the conductor which opposes the magnetic force on the electrons. Find the magnitude of the electric field which will stop further accumulation of electrons.


(d) What will be the potential difference developed across the width of the conductor due to the electron-accumulation? The appearance of a transvers emf, when a current carrying wire is placed in a magnetic field, is called Hall effect.



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A particle having a charge of 2.0 × 10–8 C and a mass of 2.0 ×10–10 g is projected with a speed of 2.0 × 103 ms–1 in a region having a uniform magnetic field of 0.10 T. The velocity is perpendicular to the field. Find the radius of the circle formed by the particle and also the time period.