Faraday’s law relates electricity and magnetism
in a special way. A changing magnetic field
induces an electric field.
Faraday's law when expressed in its differential
form, relates the rate of change of a magnetic
field to the curl of the induced electric
field.
The curl of E equals the negative partial
derivative of B with respect to time, t.
Let’s start with the magnetic field B, defined
as three x t, negative two y t, z t, from
t equals zero to infinity.
Calculate the curl of the associated electric
field E.
The curl of E equals the negative partial
derivative of B with respect to t.
We plug in B of t three x t, negative two
y t, z t.
After evaluating, we get that the curl of
E equals three x, negative two y, z.
Let’s try a more complicated example. Given
the electric field E, four y z t, y squared
t, two x y t, calculate the induced magnetic
field B.
Recall that the curl of E equals the negative
partial derivative of B with respect to t.
Solving for B, we get B equals the integral
of the negative curl of E, d t.
Calculate the curl of E. Recall that the curl
can be computed from partial derivatives.
The i component of curl E equals the partial
derivative of E sub z with respect to y minus
the partial derivative of E sub y with respect
to z, which equals two xt.
The j component of curl E equals the partial
derivative of E sub x with respect to z minus
the partial derivative of E sub z with respect
to x, which equals four yt minus two yt.
The k component of curl E equals the partial
derivative of E sub y with respect to x minus
the partial derivative of E sub x with respect
to y, which equals zero minus four yzt.
The curl of E equals two xt, two yt, negative
four yzt.
Recall that B equals the integral of the negative
curl of E d t.
We plug in the previously calculated value
for the curl of E.
Distributing the minus sign and taking the
antiderivative with respect to t yields B
equals negative x t squared, negative y t
squared, two z t squared.
