Past exam of the physics 2 paper of the University of Oxford / 2017 resit / 7 / c

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- Solution c

Solution

Let

m

be the sliding point charge's mass. Expanding the axial field of a uniformly charged annulus for

∣ x ∣ ≪ a, b

gives

E_{x} = \frac{σ}{2 ϵ _{0}} (\frac{1}{a} - \frac{1}{b}) x + O (x^{3}) = \frac{Q}{2 π ϵ _{0} ab ( a + b )} x + O (x^{3}) .

Its electric charge is

- Q

, so its electric force points toward the centre. Newton's second law becomes

m \overset{x}{¨} = - \frac{Q ^{2}}{2 π ϵ _{0} ab ( a + b )} x + O (x^{3}) .

To leading order this is simple harmonic motion, with

ω_{osc}^{2} = \frac{Q ^{2}}{2 π ϵ _{0} mab ( a + b )}, T = 2 π \frac{2 π ϵ _{0} mab ( a + b )}{Q ^{2}} .

The mass is an additional parameter, since the source supplies no numerical value. The constraint is essential: stability along the axis does not imply unconstrained three-dimensional trapping by a static electric field.

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