Now consider the magnetic flux between two lines at and , which are

The flux is

Notice that the flux is a function of time only. The spatial dependence has been

Hence, we may write

By (4.10) this is zero. In other words, the

__Example 4. .1__Consider the effect of a flow
on an initially
uniform field,
. Equation (4.10) becomes

Thus,

Look for a seperable solution to this linear equation of the form

so that (4.11) may be expressed as

Dividing by and collecting all the functions of time onto one side of the equation and all the functions of onto the other we get

Consider the -dependence first of all.

Integrating we obtain

Similarly we may integrate the dependent terms,

Hence, we obtain

In theory, we need to obtain the values for the separation constant and add together all the possible solutions. However, this example is particularly simple. Satisfying the initial condition gives

Therefore,

Thus, the flow carries in the field lines and makes the magnetic field strength grow exponentially in time.