So, let's take a look at this picture again,

let's consider a segment of wire of unit length carrying the current I,

and moving in a direction perpendicular to itself and to

magnetic field B with speed V sub wire,

which is schematically depicted right next to Melodie,

and you can see this is the section of the wire,

we have current flow here,

the drift current and then you have wire moving on the direction perpendicular to

the wire with a velocity of V sub wire and the field is pointing along Z direction.

Since the current is held constant,

the forces under conduction electrons do not cause them to accelerate,

so this is very important.

The electrical energy is not going into the electrons,

but into the source that is keeping the current constant,

so the source is taking away extra energy,

so make sure the current is kept constant.

In other words, the charge carriers are moving at a constant speed.

Note that the force on the wire is IB,

so IB times v sub wire is also the rate of

mechanical work done on the wire, you see that.

So, d_U sub mech over d_t is IBv sub wire.

Now the mechanical work done on the wire is just

equal to the work done on the current source,

and therefore the energy of the loop is a constant.

So, you can see they should be constant because rate of

total electric work and rate of mechanical work, they are the same.

So, we discussed about mechanical energy and electric energy,

electrical energy on the loop and we wonder if we're missing any other energy terms,

so Melodie are we missing any other terms?

I'm not exactly sure where the B field is coming from,

so maybe there's a term associated with that.

Exactly, in order to create B field at a constant magnitude,

we need to have a coil and that is

another energy source that we need to consider when you think about the whole system.

Now, the total force on each charge in the wire is F equals Q times parenthesis,

E plus V cross B parenthesis,

we learned that from our earlier lecture.

What did we call this force?

The Lorentz force.

Exactly Lorentz force.

This is from experiments.

Now, I want to use this to understand the rate at which work is

done and the work is defined by the force times the distance,

and the rate at which work is done is just force times the distance over time,

and if I divide time for- if I do divide the distance by time, that becomes velocity.

Therefore, velocity that force is equal to

the rate at which work is done and if I put that here,

that results in q[ v. E + v. (v

cross B)] Now in

our magnetic static- magnetostatic world we

don't have electric field because there's no net charge,

so this will be zero.

Now we also know from our vector calculus that anything,

the dot product and cross product,

if you have the same vector outside and inside the parenthesis,

this will result in zero, why?

Because these vectors and this vector will be

perpendicular and if you do dot product for two perpendicular vectors, that's zero.

So, that means if there are no electric fields,

we have only the second term which is always zero,

which means the rate at which work is done is zero.

Again, the rate at which work is done is zero,

so we shall see that changing magnetic fields produce E fields,

so our reasoning applies only to moving wires and steady magnetic fields.

Then, how is it then the principle of virtual work gives the right answer.

We still have not taken into account the total energy of the world,

we haven't included the energy of the currents that are

producing the magnetic field we start out with.

So, until here, you're puzzled because it's zero, right?

However, now we will see another term coming.