CBSE Class 12 · 2023 · ElectrostaticsLong Answer

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(a)(i) Use Gauss's law to obtain an expression for the electric field due to an infinitely long thin straight wire with uniform linear charge density

λ

. (ii) An infinitely long positively charged straight wire has a linear charge density

λ

. An electron is revolving in a circle with a constant speed

v

such that the wire passes through the centre and is perpendicular to the plane of the circle. Find the kinetic energy of the electron in terms of the magnitudes of its charge and linear charge density

λ

on the wire. (iii) Draw a graph of kinetic energy as a function of linear charge density

λ

Solution

(i) With a coaxial Gaussian cylinder (radius

r

, length

ℓ

E (2 π r ℓ) = \frac{λ ℓ}{ε _{0}} \Rightarrow E = \frac{λ}{2 π ε _{0} r}

(ii) The electrostatic force provides the centripetal force:

e E = \frac{m v ^{2}}{r} \Rightarrow e \cdot \frac{λ}{2 π ε _{0} r} = \frac{m v ^{2}}{r}

m v^{2} = \frac{e λ}{2 π ε _{0}}

K E = \frac{1}{2} m v^{2} = \frac{e λ}{4 π ε _{0}}

independent of the radius.
(iii) Since

K E \propto λ

, the graph of

K E

against

λ

is a straight line through the origin.

CBSE Class 12 Board · 2023 · Subjective

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CBSE Class 12 · 2023 · ElectrostaticsLong Answer

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