Question

A line of charge (with charge per unit length given by λ) extends from (0,0,0) along the x axis to infinity. Answer the following 2 questions about the electric field at the point (x,y,z) = (0,a,0).

1. A correct integral expression for the y component of the electric field at this point is given by which of the following? In each case the integral has limits 0 and ∞.
   1. λ/(4πε₀)∫dx a/(a² + x²)^3/2 k
   2. λ/(4πε₀)∫dx a/(a² + x²)^3/2 j
   3. λ/(4πε₀)∫dx a/(a² + x²)     j
   4. λ/(4πε₀)∫dx a/(a² + x²)^1/2k
   5. none of a-d   

                                                                   

2. What is the y component of the electric field at this point? Hint: remember (or calculate using Gauss’s law) the electric field a distance a from an infinite wire and think of a symmetry argument that allows you to answer the question without a more complicated calculation.

1. E_y = λ/(4πaε₀) j
2. E_y = λ/(2πaε₀) j
3. E_y = λ/(4πaε₀) k
4. E_y = λ/(2πaε₀) i    
5. none of a-d                                                                             

3. Which of the following is(are) true about a charged particle in a uniform, static (time-independent) magnetic field?
   1. The magnitude of the magnetic force on the particle depends only on the charge, the magnetic field and physical constants.
   2. The magnitude of the magnetic force on the particle depends on the direction of motion of the particle
   3. The work done on the particle by the magnetic force depends on the velocity of the particle.
   4. If the particle is moving, the magnetic force on the particle cannot be zero.
   5. More than one of a-d

Answer 1

1) The y-component of electric field due to an elemental charge dq = dx of infinite line charge is given by

Option b

2) By using Gauss Law, the field due to infinite line charge at y can be calculated as ;  ;

But the wire is from (0,0,0) to + infinity. Hence the field due to this semi infinite wire is half of that value as the components parallel to wire gets cancelled.

Option a

3) The magnetic force acting on a charged particle is given by ;

Option a is wrong because

The force depends on magnitude and nature of the charge, the velocity and magnetic field and also the direction of motion with respect to direction of magnetic field.

Option b is correct.

Since force is always perpendicular to velocity then work done by magnetic force is zero irrespective of its velocity.

Option c is wrong.

Option d is wrong because

If the velocity is parallel to magnetic field then angle between v and B becomes zero then magnetic force is zero. The magnetic force can be zero even if the particle is moving (if direction of motion is parallel to B).