A region of space has a magnetic flux density (vecB) given...

A region of space has a magnetic flux density (vec{B}) given by (vec{B} = (hat{a}_x + hat{a}_y)B_0) (T). Find the magnetic flux that passes through the first octant of the sphere of radius R, (0 le heta le pi/2, 0 le phi le pi/2). Assume the positive normal to the surface of the sphere, (hat{a}_R).

Transcript

00:01 Hi, in this question we are given with a magnetic field in terms of the equation.

00:06 Now we have to convert this magnetic field into spherical coordinates.

00:12 So spherical coordinates.

00:24 So this is given by so in terms of spherical coordinates b is written as r theta comma phi which is given in terms of matrix written as so this is in terms of spherical coordinates where magnetic field is written in terms of r theta and phi.

00:40 Now we have b is given by so b is equal to this sin theta cos phi plus sin theta sin phi into b naught into r cap.

00:58 So a r cap plus cos theta cos phi plus so this is one term cos theta sin phi into b naught a phi.

01:13 So this is the phi component.

01:14 Then we have plus sin phi into so sin phi plus cos phi into a theta.

01:22 So these are the theta components.

01:24 These are the phi components and these are the r components.

01:27 Now we have the surface element ds is equal to it is r square sin theta d phi d theta into a r cap.

01:38 Then we have the flux.

01:40 So magnetic flux phi is given by it is surface integration of so surface aerial integration of the magnetic field.

01:50 So we have phi is equal to double integration of b naught into so this is the amplitude of the magnetic wave.

01:57 It is sin theta into cos phi plus sin phi into sin theta multiplied by r square.

02:07 So d square ds is so taking only components of r.

02:11 So we have r square in terms of ds we have r square sin theta d phi d theta.

02:21 So we have the theta it ranges from 0 to pi by 2 and phi it ranges from 0 to pi by 2.

02:33 So the integration becomes so the integration will be phi is equal to 0 to pi by 2 0 to pi by 2.

02:42 So this is a double integration.

02:43 So b naught into r square is common.

02:46 It is taken outside...

Question

A region of space has a magnetic flux density (vec{B}) given by (vec{B} = (hat{a}_x + hat{a}_y)B_0) (T). Find the magnetic flux that passes through the first octant of the sphere of radius R, (0 le heta le pi/2, 0 le phi le pi/2). Assume the positive normal to the surface of the sphere, (hat{a}_R).

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