Q16 of 805 Page 1

Derive an expression for the velocity vC of positive ions passing undeflected through a region where crossed and uniform electric field E and magnetic field B are simultaneously present.

Draw and justify the trajectory of identical positive ions whose velocity has a magnitude less than vC.


OR


A particle of mass m and charge q is in motion at speed v parallel to a long straight conductor carrying current I as shown below.



Find magnitude and direction of electric field required so that the particle goes undeflected.

Formula: -


We know that a charge q moving with velocity v in presence of both electric and magnetic fields experiences a force (Lorentz’s Force) given by, i.e.,


… (1)


Where, E is the Electric field


B is the magnetic field


V is the velocity of the charged particle and,


q is the charge on the particle.



Let us consider that the velocity of the particle is perpendicular to both electric and magnetic fields and they are mutually perpendicular also, depicted in the figure.


We have,



and,



Substituting the values in the equation (1), we get,



Therefore,



Thus, electric and magnetic forces are in opposite directions. the charge will move in the fields undeflected only when the magnitudes of the two forces are equal. Then, total force on the charge is zero and this happens when,


qE=qvB,



If the velocity v <vC, then,


v<vC



E-vB<0


As,



So, the force is in the positive y direction so there will be acceleration in the positive y direction.



Conclusions: -


Expression for the velocity vC of positive ions passing undeflected through a region where crossed and uniform electric field E and magnetic field B are simultaneously present is,



In the second part there will be acceleration in the positive y direction


OR


Formula: -


When a charge q moves with a velocity v in vicinity of both electric and magnetic fields, it experiences a force which is called the Lorentz’s Force and mathematically is,



the magnetic field due to the wire above the x-z plane is,




So,



In order to balance this force, we need a force due to electric field as,




Also, we know that,



So,





Conclusion: -


The magnitude of the electric field is,



And the direction is in the positive y axis.


More from this chapter

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(a) An electric dipole is kept first to the left and then to the right of a negatively charged infinite plane sheet having a uniform surface charge density. The arrows P1 and P2 show the directions of its electric dipole moment in the two cases.


Identify for each case, whether the dipole is in stable or unstable equilibrium. Justify each answer.


(b) Next, the dipole is kept in a similar way (as shown), near an infinitely long straight wire having uniform negative linear charge density.



Will the dipole be in equilibrium at these two positions? Justify your answer.

15

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(a) In which bar is drift speed of free electrons greater?


(b) If the same constant current continues to flow for a long time, how will the voltage drop across A and B be affected?


Justify each answer.

17

A sinusoidal voltage of peak value 10 V is applied to a series LCR circuit in which resistance, capacitance and inductance have values of 10 Ω, 1μF and 1H respectively. Find (i) the peak voltage across the inductor at resonance (ii) quality factor of the circuit.

 18

a) What is the principle of transformer?

b) Explain how laminating the core of a transformer helps to reduce eddy current losses in it


c) Why the primary and secondary coils of a transformer are preferably wound on the same core


OR


Show that in the free oscillations of an LC circuit, the sum of energies stored in the capacitor and the inductor is constant in time.