A proton moves along the x-axis with vₓ = 1.0 x 10⁷ m/s.. As it p...

Question

A proton moves along the x-axis with vₓ = 1.0 x 10⁷ m/s.. As it passes the origin, what are the strength and direction of the magnetic field at the (x, y, z) positions (a) (1 cm, 0 cm, 0 cm), (b)(0 cm, 1 cm, 0 cm), and (c) (0 cm, -2 cm, 0 cm)?

Accepted Answer

Hi everyone. In this practice problem, we will have an alpha particle traveling along the Z axis with a constant speed. Pz equals 3.2 times 10 to the power of 7 m per second. We're being asked to determine the magnitude and the direction of the magnetic field at the point of zero centimeter comma five centimeter comma negative one centimeter when the alpha particle is at 0.0 centimeter comma zero centimeter comma negative one centimeter. The options given are a negative 2.1 times 10 to the power of negative K Tesla B negative 4.1 times 10 to the power of negative 16 I Tesla C negative 3.2 times 10 to the power of negative 14 J tesla. And lastly D 4.2 times 10 to the power of negative 14 J tesla. So to determine the magnetic field strength and the direction at a given position due to a moving alpha particle, we can use the bio Saar law where B will be equal to mu not divided by four pi multiplied by Q multiplied by V multiplied by sign of theta, all of that divided by R squared. So the formula for the magnetic field due to a moving charge can be obtained by this formula right here where B is the magnetic field factor mu knot is the per permeability of free space, which is for pi times 10 to the power of negative seven tesla meter divided by A Q is the charge of the particle, which in this case is an alpha alpha particle which is going to be equals to two multiplied by 1.6 multiplied by 10 to the power of negative 19 colum and V is the velocity factor of the particle. And R is the position factor from the charge to the point of interest. The is going to be the angle between the velocity factor and the line connecting the charge to the point of interest which in this case, since the proton is moving along the Z axis or PZ or having the velocity factor of VZ, the angle between the velocity factor and the line connecting the charge to the point will always be 90 degrees. So the will be equal to 90 degrees or pi divided by two radiant. So in this case, sine of 90 degrees will be equal to sine of pi divided by two, which will be equals to one just like. So the position factor from the alpha particle to the point it uh to the point is going to be equals to Dr which in this case is going to be equal to zero centimeter comma five centimeter comma negative one minus it brans negative one centimeter which will then uh get us the R to be equal to zero centimeter comma five centimeter comma zero centimeter and converting that into meter R will be equal to 0 m, comma 0.05 m comma m just like soap. The velocity factor of the proton P is going to be equal to 0 m per second for VX 0 m per second for Vy and for VZ 3.2 times 10 to the power of 7 m per second. So now we are ready to substitute all of the values that we have. So then B will be equal to mu not divided by four pi which is going to be four pi times 10 to the power of negative seven Tesla meter divided by a divided by four pi after this is going to be multiplied by Q which is going to be two times 1.6 times 10 to the power of negative 19 coon multiply that with uh the velocity which is going to be 2. times 10 to the power of 7 m per second multiply that with one for the sign of 90 degrees and then all of that is going to be divided by A R which is going to be 0.05 m squared. So since uh there's only one component for the R and one component for the fee, the R and the fee are just going to be those values for the other component. But if that's not the case, then we will have to find a magnitude for both R and P. But calculating this B will then be equal to 4.1 times 10 to the power of negative 16 tesla. And using the right hand rule to find the direction of the magnetic field, we want to point our time in the direction of the velocity factor. And the magnetic field factor B is going to be perpendicular to the plane of R and P and the point in the same direction that our index fingers point in the rational case, the fingers point is going to be along the negative K direction. And because of that, the magnetic field is going to then be pointing to the negative I direction. So B equals negative 4.1 times 10 to the power of negative 16 I tesla is going to be the answer to this particular practice problem which will be corresponding to option B in our answer choices. So option B will be the answer to this particular practice problem. And that will be it for this video. If you guys still have any sort of confusion, please feel free to check out our adolescent videos on similar topics available on our website. But other than that, that will be it for this one. Thank you.

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A proton moves along the x-axis with vₓ = 1.0 x 10⁷ m/s.. As it passes the origin, what are the strength and direction of the magnetic field at the (x, y, z) positions (a) (1 cm, 0 cm, 0 cm), (b)(0 cm, 1 cm, 0 cm), and (c) (0 cm, -2 cm, 0 cm)?

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