The magnetic field of an electromagnetic wave is described...

The magnetic field of an electromagnetic wave is described by ⃗ B(z, t) = 10−6 sin(105z − ωt)(−ˆx) T Determine the electric field completely. b: If you examined the magnetic field at a fixed time, what is the distance from one peak to another peak (in the same direction)? c: Determine the Poynting vector as a function of z and t. d: Determine the spatial gradient of the magnetic field ( ∂B ∂z ) at z = 10−4 m and t = 10−13 s. e: Determine the maximum time rate of change of the electric field (which would be the same at any point in space

Transcript

00:01 Hello students, in this question we are given that an em wave is traveling in vacuum along the propagating along x direction and which has a wavelength lambda which is equal to 497 nanometers and we are given that the equation for the wave oscillation is given to us as b equal to b1 sine of k z minus omega i'll cap plus jt so first of all we have to in the first part of the question we have to calculate the wave number that is denoted by letter k that is the wave number we have to calculate this wave number which is equal to k is equal to 2 pi divided by the lambda which is the wavelength and in order to putting the values in this equation that is 2 pi divided by the wavelength that is 497 multiplied by 10 to the power minus 9 meters are k that is the wave number is coming out to be 1 .26 multiplied by 10 to the power cell.

01:03 So this is the wave number for this wave.

01:06 Now, in the second part of the question, we have to calculate z max, that is the distance along the positive x -axis to the positive position where the magnitude of magnetic field is maximum at time t is equal to 0.

01:21 We have to calculate the x max now in the second part of the question we have to calculate the z max at t is equal to zero so at t is equal to zero sign of k z minus that is the phase term is equal to one this means that k z minus omega t would be equal to pi by two because and as t is equal to 0 at we are given we have to calculate as t is equal to 0 that is the omega t terms gets cancelled that is this terms get cancelled so k z is equal to pi by 2 pi divided by 2 so z would be equal to pi divided by 2 k so this is the term this is the value of z max and we can calculate the value of z max by putting the value that would be equal to 5 divided by 2 and k is the wave number which we calculated earlier as 1 .26 multiplied by 10 to the power 7 and her z max is coming out to be 1 .24 multiplied by 10 to the power minus 7 so this is the set mix now in the next part of the question we have to calculate are the magnitude of maximum electric field that is we have to calculate the e max we have to calculate emax and emax is equal to b max that is the coefficient of the maximum magnetic field multiplied by the c that is the speed of light c is speed of light speed of light in vacuum that is equal to 3 .0 multiplied by 10 to the power 8 meters so first of all in order to calculate e max we have to calculate the b max that is equal to b max is equal to root 2 times b1 that is as in our equation we are given that we are given that b1 is equal to b1 multiplied by i cap plus j cap that it is oscillation is oscillation is along the z direction so b max would be equal to root 2 multiplied by b1 first of all we calculate b1 that is root 2 multiplied by b max so that would be equal to 1 .414 multiplied by b1 the magnitude of b1 is given to us in the quotient as 11 .3 multiplied by 10 to the power minus 6 tesla and our b max is equivalently coming out to be 15 .9 multiplied by 10 to the power minus 6 tesla that is the value of b max and now we put this.

04:30 This value of b max in this is in in emacs equation in order to calculate the magnitude of em max so the magnitude of em max would be coming out to be em max would be coming out to be 15 .9 multiplied by 10 to the power minus 6 multiplied by the speed of light in vacuum that is 3 .0 multiplied by 10 to the power 8 and uh e max is subsequently coming out to the 4 .79 multiplied by 10 to the power 3 newton per cooler...

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