# Prove that magnetic forces do no work.

By definition, work is the scalar product of the force and displacement that occurred during the action of the force:

`W = vecF*Deltavecx` . The force, however, is not necessarily the cause of this displacement.

The scalar product of any two vectors equals the product of the magnitudes of the vectors and the cosine of the angle between them. So, if the force makes an angle `theta`

with the displacement vector, its work can be calculated as

`W = F*Deltax*costheta` . (In this expression, the force and the angle are assumed to be constant, but the expression for work and the reasoning that follows can be generalized for the varying force and angles as well - see the reference link.)

From this expression, notice that the work done by a given force can be zero if

1) there is no displacement. You could be exerting a lot of effort pushing on a heavy boulder, but if it does not move, the force with which you are pushing does no work.

2) the cosine of the angle is 0: `cos(theta) = 0`

This is the case when the angle equals 90 degrees, which means that the force and the displacement vectors are perpendicular. This leads to the conclusion that the forces perpendicular to the displacement do no work. Since velocity always has the same direction as displacement (`vecv = Deltavecx/(Deltat)` ), this also means that the forces perpendicular to the velocity do no work. For example, in a circular motion of a weight attached to a string, the tension force does no work, because it is directed towards the center of the circle, perpendicular to the velocity (and displacement), which is tangent to the circle.

Now let's consider magnetic force. The magnetic force on a moving electric charge q equals

`vecF = q*vecv xx vecB` , the value of the charge times the vector product of the velocity and the magnetic field. The result of the vector product, by definition, is a vector perpendicular to the both vectors being multiplied. So the magnetic force is always perpendicular to both velocity and magnetic field. Similarly, the magnetic forces are always perpendicular to the direction of the electric current.

As discussed above, the forces perpendicular to the velocity do no work. Therefore, magnetic forces, which are perpendicular to the velocity and displacement of charges, do no work.

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Use the Lorentz force on a charge `Q` , when `Q` moves an amount `dl` . Then we also know that work is the dot product of force and distance.

`dW_(mag)=F_(mag)*dl=Q(v xx B)*dl`

But `dl=v dt`

`dW_(mag)=Q(v xx B)*v dt`

Let `v xx B=v_(_|_ )` where `v_(_|_)` is a vector that is perpendicular to both `v` and `B` .

Then,

`dW_(mag)=Q(v xx B)*v dt=Q(v_(_|_)*v)dt`

The dot product of two perpendicular vectors is zero. Therefore,

`dW_(mag)=Q(0)dt=0`

Magnetic forces do no work.