During transcranial magnetic stimulation (TMS) treatment, a magnetic field typically of magnitude 0.40 T 0.40 T is produced in the brain using external coils. During the treatment, the current in the coils (and hence the magnetic field in the brain) rises from zero to its peak in about 75 μs . 75 μs. Assume that the magnetic field is uniform over a circular area of diameter 2.2 cm 2.2 cm inside the brain. What is the magnitude of the average induced emf | E | |E| around this area in the brain during the treatment?

Since there is a variation in the current flow on the coil , it means that the magnetic field in the brain would also vary causing an emf to be induced

and from the question the current varies from zero to peak so would the magnetic field vary from zero to peak(i.e from 0 to 0.40T)

This induced emf is mathematically represented as

[tex]e = A \frac{dB}{dt}[/tex]

Where dt = [tex]t_1 -t_0[/tex]

where [tex]t_0 =0 s[/tex]

and [tex]t_1 = 75\mu s= 75*10^{-6}s[/tex]

So dt [tex]= 75*10^{-6}s[/tex]

A is the area of the coil mathematically evaluated as

A = [tex]\pi r^2[/tex]

Where [tex]r= \frac{d}{2}[/tex]

Substituting 2.2cm = [tex]\frac{2.2}{100} = 2.2*10^{-2}m[/tex] for d

[tex]r=1.1 *10^{-2}m[/tex]

Then [tex]A = 3.142 * (1.1*10^{-2})^2[/tex]

[tex]=3.80*10^{-4}m^2[/tex]

Now [tex]e = 3.80*10^{-4} \frac{0.40 -0}{75*10^{-6}}[/tex]

[tex]e= 2.02V[/tex]