Force on an electron due to a time-varying magnetic flux density through its circular orbit An electron orbits a proton in a circle of radius R. If a spatially uniform magnetic flux density B threading the orbit is increased at a rate dB/dt, what is the tangential force on the electron arising from the induced electric field?

Introduction / Context:
A time-varying magnetic field induces a circulating electric field according to Faraday’s law. A charge on a circular path then experiences a tangential electric force. This concept underlies electromagnetic induction, eddy currents, and betatron acceleration.

Given Data / Assumptions:

• Orbit is circular with radius R and threaded by a uniform B(t).
• Magnetic flux through the orbit is Φ = π R^2 B.
• Rate of change dB/dt is spatially uniform across the loop.
• Electron charge magnitude is e.

Concept / Approach:

Faraday’s law in integral form gives the induced emf around a closed path: emf = − dΦ/dt. For a circle, the induced tangential electric field E_t is constant around the circumference, so emf = ∮ E · dl = E_t * 2π R. Substituting Φ = π R^2 B relates E_t to dB/dt.

Step-by-Step Solution:

Verification / Alternative check:

Dimensional check: [e]·[R]·[dB/dt] → C·m·(T/s) = N, which is correct for force. The direction is azimuthal, consistent with Lenz’s law (sign depends on increasing or decreasing B).

Why Other Options Are Wrong:

(b) misses the factor 1/2; (c) uses B instead of dB/dt; (d) has wrong dependence on R; (e) magnetic fields themselves do no work, but the induced electric field does.

Common Pitfalls:

Confusing the roles of B and the induced E, or forgetting the circumference factor 2πR in Faraday’s integral law.

Final Answer:

(e R / 2) * (dB/dt)