Question

A time-dependent but otherwise uniform magnetic field of magnitude Bo(t) is confined in a cylindrical region of radius 6.5 cm. Initially the magnetic field in the region is pointed out of the page and has a magnitude of 3.5 T, but it is decreasing at a rate of 29.5 G/s. Due to the changing magnetic field, an electric field will be induced in this space which causes the acceleration of charges in the region.

Required:
What is the direction of acceleration of a proton placed in at the point 1.5 cm from the center?

Answer 1

The acceleration is
`a = 9197.11 m/s^2`

Step-by-step explanation:

From the question we are told that

The radius of the cylinder is
`r = 6.5 \ cm = (6.5)/(100) = 0.065 m`

The magnitude of the magnetic field is
`B_i = 3.5T`

The rate of decrease is
`(dB)/(dt) = 29.5 \ G/s = (29.5)/(10000) = 29.5 * 10 ^(-4) T/s`

The distance from the center is
`D = 1.5\ cm = (1.5)/(10)=0.15m`

Faraday's law of induction states mathematically that

`\epsilon = (d \o)/(dt)`

`\epsilon = \int\limits {E} \, dl`

Where
`\epsilon` is the induced emf

E is the magnitude of the electric field

`dl` i the change in length

`d \o` is the change in magnetic flux which is mathematically represented as

`\o = BA`

substituting this into the above equation

`\int\limits {E} \, dl = (d(BA))/(dt )`

`E l = A (dB)/(dt)`

Where l is the circumference of the circular loop formed in the cylinder which is mathematically represented as

`l = 2\pi r`

And A is the area of the circular loop formed which is mathematically represented as

`A = \pi r^2`

So

`E (2 \pi r ) = (\pi r^2 ) (dB)/(dt)`

`E = (r)/(2) (dB)/(dt)`

Substituting value

`E = (0.065)/(2) * 29.5 *10^(-4)`

`= 9.588*10^(-5) \ V/m`

Generally acceleration is mathematically represented as

`a = (F)/(m)`

Now F is the electric force which is mathematically represented as

`F = qE`

Substituting this into the question

`a = (qE )/(m )`

Where q is the charge on the proton with a constant value of

`q = 1.60 *10^ {-19 } C`

and m is the mass of the proton with a constant value of

`m = 1.67 *10^(-31) kg`

Substituting values

`a = (1.60*10^(-19) * 9.588 *10^(-5))/(1.67 *10^(-27))`

`a = 9197.11 m/s^2`