Magnetic Field Induced by Nonuniform Electric Flux

[Badre]

Homework Statement

During the design of a new doppler radar unit, you have two circular plates with radius R = 3 cm and a plate separation of d = 5 mm. The magnitude of the electric field between the plates is given as:

E = (700 V/(m sec2))*(1-r/Rp)*t 2

where Rp is the radius of the plates, t is the time in sec and r is the distance from the axis of the plates (for r < Rp ).

a) What is amplitude of the induced magnetic field a distance 1.5 cm from the center axis joining the plates at time t = 2.9 sec?

Homework Equations

∫B.ds = $\mu_{0}$$\epsilon_{0}$ d$\Phi$e/dt

The Attempt at a Solution

I did this problem as I worked out previous ones, taking the time derivative of the electric field to find the changing electric flux.

B=$\mu_{0}$$\epsilon_{0}$ pi*r^2/2*pi*r * dE/dt
B=$\mu_{0}$$\epsilon_{0}$ (.015/2) * 2*700t(1-.015/.03)
B = 1.69E-16

This answer is rejected by the software. I'm at a loss for what I'm missing here. I don't understand why I'm given the separation between the plates, I suspect that may be involved in the solution but I'm not seeing how. Thanks for any help.

 

[collinsmark]

Homework Statement

During the design of a new doppler radar unit, you have two circular plates with radius R = 3 cm and a plate separation of d = 5 mm. The magnitude of the electric field between the plates is given as:

E = (700 V/(m sec2))*(1-r/Rp)*t 2

where Rp is the radius of the plates, t is the time in sec and r is the distance from the axis of the plates (for r < Rp ).

a) What is amplitude of the induced magnetic field a distance 1.5 cm from the center axis joining the plates at time t = 2.9 sec?

Homework Equations

∫B.ds = $\mu_{0}$$\epsilon_{0}$ d$\Phi$e/dt

The Attempt at a Solution

I did this problem as I worked out previous ones, taking the time derivative of the electric field to find the changing electric flux.

B=$\mu_{0}$$\epsilon_{0}$ pi*r^2/2*pi*r * dE/dt

Okay, I think I see what you are doing here (it took me awhile to figure out your above result).

When you tried to calculate the flux, you just multiplied the electric field E by the area, πr². That would work if the electric field was [STRIKE]constant[/STRIKE] uniform within the entire circle defined by r (it can be time changing, but at a particular point in time it must be uniform across the surface defined within r). But it's not [STRIKE]constant[/STRIKE] uniform for this problem. (Although due to symmetry, the magnetic field strength is [STRIKE]constant[/STRIKE] uniform around the perimeter 2πr, which is why the ∫B·ds = B[2πr] part worked okay.)

You're going to have to do an integration, one way or the other. You have two choices. You can calculate the flux directly via

$$\Phi_E = \int_S \vec E \cdot \vec{dA},$$

and then take the time derivative to find $\frac{\partial \Phi_E}{\partial t}$. Or, you can take the time derivative first to obtain $\frac{\partial E}{\partial t}$, and then evaluate

$$\frac{ \partial \Phi_E}{\partial t} = \int_S \frac{\partial \vec E}{\partial t} \cdot \vec{dA}.$$

Either way works fine.

It may be useful to note that in polar coordinates, $dA = r \ dr \ d\theta$.

 

[Badre]

Thanks! Haven't had to do a double integral before now.