Calculate the magnitude of the magnetic field at point P due to the current in the semicircular section of wire shown in the figure . (Hint: Does the current in the long, straight section of the wire produce any field at P?)Express your answer in terms of the variables I, R and appropriate constants.
B) Find the direction of the magnetic field at point P: into the page or out of the page?
2 Answers
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(A) Integrate around the curve, from 0 to π radians, to sum up the total at P. Ignore the straight part, since obviously that doesn’t contribute when the radial distance from the straight part is zero.
I’ll use the Biot-Savart law instead of Ampere’s law, since there’s isn’t symmetry to simplify the line integral.
dB = (µ0/4π) I dsxR / R^3 where x means cross product
ds is the wire element vector pointing tangent to the wire.
So I ds is a differential current element.
r is a vector pointing toward P.
Since s and r are perpendicular all along the semi-circle, the cross product is just ds * r.
We rewrite dB as
(µ0/4π) I ds / R²
I and R are constant. So we’re really just integrating ds over the wire curve from 0 to π.
So all we’re finding by integrating is the length of the wire that’s curved. That’s just πR.
So B = µ0 I / 4R
(B) Using the right-hand rule, it should be pretty obvious.
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Calculate the magnitude of the magnetic field……………?
Calculate the magnitude of the magnetic field at point P due to the current in the semicircular section of wire shown in the figure . (Hint: Does the current in the long, straight section of the wire produce any field at P?)Express your answer in terms of the variables I, R and appropriate…
