Question

A particle of charge q is moving with a velocity `vec v` at a distance ‘d’ from a long straight wire carrying a current ‘I’ as shown in figure. At this instant, it is subjected to a uniform electric field `vec E` such that the particle keeps moving undeviated. In terms of unit vectors `hat i`, `hat j` and `hat k`, find -

1. the magnetic field `vec B`,
2. the magnetic force `vec F_m`, and
3. the electric field `vec E` acting on the charge.

Answer 1

a. Magnetic field `vec B`:

According to Ampere’s law

B = `(mu_0I)/(2 pi d)`

The direction is determined by right hand rule, and the magnetic field direction is along the negative direction.

`vec B = (mu_0 I)/(2 pi d) hat j`

b. Magnetic force `vec F_m`:

charge is many along the x direction,

So, `vec v xx vec B = (v vec i) xx ((mu_0 I)/(2 pi d) hat j)`

`vec F_B = qv(-(mu_0 I)/(2 pi d) hat k)`

So the direction of the magnetic force is negative.

c. Electric field `vec E`:

The charge is moving undeviated:

`vec F_E + vec F_B` = 0

`vec F_E = q vec E`

⇒ `q vec E = -(-(qvmu_0I)/(2 pi d)hat k)`

`vec E = (v mu_0 I)/(2 pi d)hat k`