Nonuniform electric field. In Fig. 32-30, an electric field is directed out of the page within a

Nonuniform electric field. In Fig. 32-30, an electric field...

Key Concepts

Maxwell's Equations

Maxwell's equations are the fundamental laws governing electricity and magnetism. In this context, the Ampère-Maxwell law is especially important as it relates a time-varying electric field to an induced magnetic field. This equation extends Ampère's law by including the displacement current, allowing for the description of magnetic fields generated not only by conduction currents but also by changing electric fields.

Displacement Current

The displacement current is a term added to Ampère's law that accounts for the contribution of a changing electric field in regions where no actual charge flow exists. It is essential for explaining how time-varying electric fields can produce magnetic fields, maintaining the continuity of current in Maxwell's equations and ensuring consistency with the principle of conservation of charge.

Circular Symmetry

Circular symmetry plays a crucial role when dealing with induced magnetic fields in systems with radially dependent electric fields. When the electric field within a circular region varies with radius and time, symmetry allows us to deduce that the induced magnetic field lines form concentric circles. This simplifies the application of the integral form of Maxwell's equations in determining the field's magnitude at different radial distances.

Integral Approach in Field Calculations

The integral form of Maxwell's equations, such as the Ampère-Maxwell law, is used to calculate the induced magnetic field by integrating along closed loops. By considering the symmetry of the problem, one can integrate over paths (usually circles) where the magnetic field is constant. This integration relates the rate of change of the electric flux through the area enclosed by the loop to the magnetic field circulating around it.

Transcript

0:00 Hi there.

00:01 So for this problem, we have the figure that is shown in here.

00:06 An electric field is directed out of the screen within a circular region and the radius art is equal to 3 centimeters.

00:20 That emeters is 0 .03 meters.

00:26 And the field magnitude is also given.

00:31 So the magnitude for the magnitude for for the electric field, it is 0 .5 balls per meter per second.

00:48 And this times 1 minus the ratio between r and the radius of the circular region times the time t.

01:04 And art, lowercase r, is the distance that we know is equal or less than this, the radius of this circular region.

01:21 Now, what we need to determine is the magnitude of the induced magnetic field at some radial distances.

01:29 So for the first part a, we need to determine the magnitude of the magnetic field.

01:35 When the distance art is equal to 2 centimeters.

01:48 Now, here the enclosed electric flux is found by integrating.

01:57 So we will find that the integrating the electric flux is equal to the integral from 0 to r since r is less than the radius of this circular region.

02:17 So we will have that this is equal to the electric field times 2 pi the radius times the differential in the radius.

02:32 Now we can take out of this everything that is common and we put the expression that we have for this.

02:40 The electric field in here.

02:43 So taking out everything that is common, we can take out the time, we can take out the magnitude that is given, 0 .5 balls per meter per second.

02:58 And also the 2 pi term, it is constant.

03:05 And we will have then the integral from 0 to art of 1 minus r over capital art times art in the differential.

03:21 So in here what we are going to have, let me do the integral separately, we will have, we multiply this by each of these terms, so we will have in here that that is from 0 to art of art minus r square over art.

03:50 This, so the first integral, we know that the integral of art is r to square over two, and the integral of r squared is r to 3 over 3.

04:10 3 over 3.

04:12 And these 3 times the radius.

04:14 So we now evaluate this from 0 to art.

04:20 And this will give us simply that expression because at 0 is just 0.

04:32 Now we will find then that the flips of electric field is equal to t times pi times the radius in here that we obtain.

05:06 Now what we are going to obtain after taking the derivative with respect to time is that the product between the magnetic field and 2 pi art it's equal to epsilon sub 0 times mu of 0 pi pi pi, the expression that we obtain.

05:42 Remember that to obtain this we need to take the derivative with respect to time for the flops that we just have obtained.

05:52 So we will obtain just simply that this is one and we obtain that expression from it.

05:59 So it is 1 over 2 the radius squared minus the radius to the cube three times capital radius...

Please give Ace some feedback