How Is the Displacement Current Calculated in a Charging Capacitor?

[StephenDoty]

A parallel-plate capacitor, C=5 mircoF, has closely spaced circular plates of radius R = .1m.

How much displacement current is encircled in a loop of r=.05m that is centered on the axis between the plates when a uniform electric field between these plates changes with time at the rate of 9*10^12 V/ms during the charging process? What is the magnitude of the induced magnetic field at r=.05m?

Dissipative current = $$\epsilon$$_0 * d$$\phi$$_e/dt
or $$\epsilon$$_0 * A *dE/dt

And for the magnetic field B =($$\epsilon$$_0 * $$\mu$$_0 * A *dE/dt)/ 2*pi*r

Now is A equal to pi*r^2 where r is equal to .05m since this is the area that the flux is going through or does A equal the ratio of the areas, r^2/R^2, since there is flux through the whole .1m radius?

Thanks.
Stephen

 

[jbsteele]

The area A is equal to pi*r^2, where r is the radius of the loop (0.05 m). So the displacement current is:I = \epsilon_0 * \pi *(0.05 m)^2 * 9 * 10^{12} V/ms = 4.4 * 10^{-6} AThe magnitude of the induced magnetic field at r=0.05m is:B = (\epsilon_0 * \mu_0 * \pi *(0.05 m)^2 * 9 * 10^{12} V/ms) / (2 * \pi * 0.05 m) = 8.8 * 10^{-5} T

 

[thomate1]

Thank you for your question. In this scenario, the displacement current is the rate of change of the electric flux through the loop, and it is given by the formula \epsilon_0 * A *dE/dt, where A is the area of the loop. Since the loop is centered on the axis between the plates, the area A is equal to pi*r^2, where r is the distance from the center of the loop to the edge of the plates (in this case, r=0.05m). This is because the electric field is uniform between the plates and the flux is going through the entire area of the loop.

Now, for the induced magnetic field, we can use the formula B =(\epsilon_0 * \mu_0 * A *dE/dt)/ 2*pi*r. Again, A is equal to pi*r^2, and the electric field is changing at a rate of 9*10^12 V/ms, so we can plug in these values to get the magnitude of the induced magnetic field at r=0.05m.

I hope this helps clarify your understanding of induced magnetic fields and the calculations involved. If you have any further questions, please don't hesitate to ask. Keep up the scientific curiosity!