Aim for FivePhysics C: E&M

Mathematical routines (MR)

Sphere with a charge density that grows with radius

  • Unit 8
  • 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 solid, nonconducting sphere of radius R has a nonuniform volume charge density ρ(r)=Ar\rho(r) = Ar, where r is the distance from the center of the sphere and A is a positive constant. There are no other charges nearby.

Suggested time: 22 minutes

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Part (a)

2 points

Derive an expression for the total charge Q of the sphere in terms of A, R, and physical constants, as appropriate.

Type math plainly, like x^2, sqrt(x) or (x+1)/(x−1).

0 / 2,500 characters

Part (b)

2 points

Use Gauss’s law to derive an expression for the magnitude of the electric field at a distance r from the center, where r < R. Express your answer in terms of A, r, and physical constants, as appropriate.

Type math plainly, like x^2, sqrt(x) or (x+1)/(x−1).

0 / 2,500 characters

Part (c)

2 points

The sphere has R = 0.10 m and A = 2.0 × 10⁻³ C/m⁴. Calculate (i) the total charge Q and (ii) the magnitude of the electric field at the surface of the sphere.

Type math plainly, like x^2, sqrt(x) or (x+1)/(x−1).

0 / 2,500 characters

Part (d)

2 points

You can’t draw here, so describe the graph you would sketch of the electric field magnitude E as a function of r, from r = 0 to r = 3R. Describe the shape in each region and how the two regions meet at r = R.

Type math plainly, like x^2, sqrt(x) or (x+1)/(x−1).

0 / 2,500 characters

Part (e)

2 points

A student claims that the magnitude of the electric field at r = R/2 is equal to the magnitude of the electric field at r = 2R. Do you agree or disagree? Justify your answer using your results from earlier parts.

Type math plainly, like x^2, sqrt(x) or (x+1)/(x−1).

0 / 2,500 characters

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