A charged particle moves with a constant velocity v = 3 i - 2 j + k m s ⁻¹ through a region containing both a uniform electric field and a uniform magnetic field. It passes through the region undeflected. If the magnetic field is B = i + 2 j - 2 k T and the electric field is E = -2 i - 7 j - k V m ⁻¹ , the value of is ____.

Step-by-step solution

For a charged particle to pass undeflected through crossed electric and magnetic fields, the net Lorentz force must be zero. F _ net = q E + q( v B ) = 0 E = -( v B ) First, calculate the cross product v B : v B = (3 i - 2 j + k ) ( i + 2 j - 2 k ) Using the determinant method: v B = i [(-2)(-2) - (1)(2)] - j [(3)(-2) - (1)(1)] + k [(3)(2) - (-2)(1)] v B = i [4 - 2] - j [-6 - 1] + k [6 + 2] v B = 2 i + 7 j + 8 k Now, find the required electric field: E = -(2 i + 7 j + 8 k ) = -2 i - 7 j - 8 k We are given that E =