A beam of positively charged particles is moving with a velocity of 4 10^5 ~m~s ⁻¹ along the positive x-axis. It enters a region containing a uniform electric field of 8 10^4 ~V~m ⁻¹ directed along the positive y-axis. To ensure that the particles pass through this region without any deviation, a uniform magnetic field must be applied. What should be the magnitude and direction of this magnetic field?

Step-by-step solution

For the positively charged particles to pass through the region without any deviation, the net force acting on them must be zero. This means the electric force and the magnetic force must be equal and opposite. The Lorentz force equation is given by: F = q E + q( v B ) = 0 E = -( v B ) Given: Velocity, v = 4 10^5 i m s ⁻¹ Electric field, E = 8 10^4 j V m ⁻¹ Substituting these values into the equation: 8 10^4 j = -(4 10^5 i B ) For the cross product i B to result in a - j direction, B must be in the k direction (sin