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Physics / Chapter 28: Magnetic Fields / Section 28-5

A Circulating Charged Particle — Question 2

Section 28-5 · Physics pages 4554–4555

Question 2 An accelerating voltage and orbit radius determine mass per unit charge Positive ions start from rest and cross potential magnitude U=2000 V before entering a uniform .200 T field perpendicular to their velocity. Their measured circular radius is .0500 m. Use nonrelativistic mechanics. r⃗ B into the page ions accelerated through U Schematic; labels, not drawing lengths, specify the data. (a) Derive and evaluate m/q without first calculating the speed. (b) If q=1.60 × 10−19 C, find m and v, and verify the original energy equation.
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Question 2 - Solution An accelerating voltage and orbit radius determine mass per unit charge r⃗ B into the page ions accelerated through U (a) Method and calculation qU = 1 2mv2, qBr = mv ⇒m q = B2r2 2U . m/q = 2.50 × 10−8 kg/C. (b) Interpretation and further calculation m = 4.00 × 10−27 kg, v = Br/(m/q) = 4.00 × 105 m/s. 1 2mv2 = 3.20 × 10−16 J = qU. Physical check The radius depends on both momentum and charge; the accelerating stage supplies the energy constraint needed for inference.

Question 2

An accelerating voltage and orbit radius determine mass per unit charge

Positive ions start from rest and cross potential magnitude U=2000 V before entering a uniform .200 T field perpendicular to their velocity. Their measured circular radius is .0500 m. Use nonrelativistic mechanics.

Diagram shown in the original SVG worksheet.

  1. Derive and evaluate m/q without first calculating the speed.

  2. If q=1.60×10−191.60\times10^{-19} C, find m and v, and verify the original energy equation.

Question 2 - Solution

An accelerating voltage and orbit radius determine mass per unit charge

Diagram shown in the original SVG worksheet.

(a) Method and calculation

qU=12mv2,qBr=mv⇒mq=B2r22U.qU=\tfrac12mv^2,\quad qBr=mv\Rightarrow\frac mq=\frac{B^2r^2}{2U}.m/q=2.50×10−8kg/C.m/q=2.50\times10^{-8}\,\mathrm{kg/C}.

(b) Interpretation and further calculation

m=4.00×10−27kg,v=Br/(m/q)=4.00×105m/s.m=4.00\times10^{-27}\,\mathrm{kg},\quad v=Br/(m/q)=4.00\times10^5\,\mathrm{m/s}.12mv2=3.20×10−16J=qU.\tfrac12mv^2=3.20\times10^{-16}\,\mathrm J=qU.

Physical check

The radius depends on both momentum and charge; the accelerating stage supplies the energy constraint needed for inference.

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