A cubical region of space is filled with some uniform electric and magnetic fields. An electron enters the cube across one of its faces with velocity `vecv` and a positron enters via opposite face with velocity `-vecv`. At this instant,

Options

A. the electric forces on both the particles cause identical accelerations.
B. the magnetic forces on both the particles cause equal accelerations.
C. both particles gain or loose energy at the same rate.
D. the motion of the centre of mass (CM) is determined by B alone.

Answer

b, c, d

Explanation

Ife is the charge and m is the mass of an electron, then force experienced by electron in an external electric field will be:
` ( vec F_(e))_(E) = - e vec E (" opposite to " vec E)`
` :. ` Acceleration of electron in electric field,
`(vec a_(e))_(E) = ((vecF_(e))_(E))/m = (- e vec E)/m` ...(i)
For a positron, charge is equal and opposite to electron and mass is name. So, the force experienced by the position will be:
`(vecF_(p))_(E) = ((vecF_(p))_(E))/m = (e vec E)/m` ....(ii)
From (i) and (ii) , we can say that
` (vec a_(e))_(E) ne ( vec a_(p))_(E)`
If v is the velocity of electron, then magnetic force experienced by the electron in magnetic field will be
`(vecF_(e))_(M) = - e (vec vxx vec B)`
` :. ` Acceleration of electron in magnetic field,
`(veca_(e))_(M) = ((vecF_(e))_(E))/m = (- e(vec v xx vecB))/m ` ...(iii)
Force experienced by the positron in external magnetic field:
`(vecF_(P))_(M) = e ( - vecv xx vec B)" " [|v|" is same"]`
` :. ` Acceleration of positron in magnetic field
`(veca_(p))_(M) = ((vecF_(p))_(M))/m = (- e( vec v xx vec B))/m ` ....(iv) From (iii) and (iv), we can say that
` ( vec a_(e))_(M) = ( vec a_(p))_(M)`
The electron and positron have the same acceleration due to magnetic field. They also have the same mass as well as the same charge.
There is no work done by magnetic field on both the charges. Electric force acting on them is same in magnitude but in opposite directions so energy sained by both the particles is going to remain same because everything else is same for both the particles except the opposite direction of force. So, they will gain or lose energy at the same rate. For the electron-positron pair, the net electric force will be:
`(vecF_(e))_(E) + (vecF_(p))_(E) = - e vec E + vecE = 0 `
Net magnetic force will be :
`(vecF_(e))_(M) + (vecF_(p))_(M) = - e ( vec v xx vecB) + [ - e ( vec v xx vec B)]`
` = - 2e ( vec v xx vec B)`
Since, the net force on the pair is dependent on `vec B` alone, so the motion of CM will be determined by `vec B` alone.
Hence, options (b), (c) and (d) are correct.