A charged particle moving with velocity v = (2 i + 3 j + c k ) m/s enters a uniform magnetic field B = (4 i + 3 j ) T . The magnetic force acting on the particle is measured to be F = (-6 i + 8 j - 2 k ) N . What is the value of the unknown velocity component c ?

Step-by-step solution

The magnetic force on a moving charge is given by F = q( v B ) . A fundamental property of the cross product is that the resulting vector is always perpendicular to both of the original vectors. Therefore, the magnetic force F must be perpendicular to the velocity v . For two perpendicular vectors, their dot product is zero: F v = 0 Substitute the given vectors into the dot product: (-6 i + 8 j - 2 k ) (2 i + 3 j + c k ) = 0 Calculate the dot product by multiplying corresponding components: (-6)(2) + (8)(3) + (-2)(