Mathematical routines (MR)
Magnetic field at the center of two semicircular arcs
- Unit 12
- 10 points
- About 22 minutes
You can use a calculator on this question, just like on exam day.
A multipart problem where you use math, including calculus, to analyze a physical situation. You draw or sketch a representation such as a diagram or a graph, derive an equation using letters, calculate a numerical answer with units, and make a claim or prediction and justify it with physics reasoning. On the exam: Question 1 of 4. Section II has 4 questions in 95 minutes (50% of the exam score), one of each type in this order; calculator and equation sheet allowed. The CED suggests 20–25 minutes.
The question
A wire loop lies in the plane of the page and carries a steady current I. It is made of two semicircular arcs that share the same center point P, joined by two straight segments that lie along a horizontal line through P. The inner arc has radius a, the outer arc has radius b, and both arcs are above the line. The current goes clockwise along the outer arc (from its left end, over the top, to its right end), then along the right straight segment toward P, then counterclockwise along the inner arc (from its right end, over the top, to its left end), and then along the left straight segment away from P, back to the start.
Suggested time: 22 minutes
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Part (a)
1 pointExplain why the straight segments of the loop produce no magnetic field at point P.
Type math plainly, like x^2, sqrt(x) or (x+1)/(x−1).
0 / 2,500 characters
Part (b)
2 pointsStarting from the Biot–Savart law, derive an expression for the magnitude of the magnetic field at P due to the inner arc alone.
Type math plainly, like x^2, sqrt(x) or (x+1)/(x−1).
0 / 2,500 characters
Part (c)
2 pointsDerive an expression for the magnitude of the net magnetic field at P, and state its direction.
Type math plainly, like x^2, sqrt(x) or (x+1)/(x−1).
0 / 2,500 characters
Part (d)
1 pointThe current is I = 4.0 A, a = 0.050 m, and b = 0.10 m. Calculate the magnitude of the net magnetic field at P.
Type math plainly, like x^2, sqrt(x) or (x+1)/(x−1).
0 / 2,500 characters
Part (e)
2 pointsA proton passes through P moving to the right (along the horizontal line) with speed 2.0 × 10⁵ m/s. Calculate the magnitude of the magnetic force on the proton at P, and state its direction.
Type math plainly, like x^2, sqrt(x) or (x+1)/(x−1).
0 / 2,500 characters
Part (f)
2 pointsA student claims that if the outer arc were made very large (b much greater than a), with the same current, the field at P would be about twice its present value. Do you agree or disagree? Justify your answer.
Type math plainly, like x^2, sqrt(x) or (x+1)/(x−1).
0 / 2,500 characters
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