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A particle of mass m and positive charge q is projected towards an infinity long line of charge from a distance r the velocity v makes 30 degree with normal to the line charge the minimum distance of approach of charge particle with line of charge

Asked by ayushsahuopp | 21 Jan, 2020, 10:03: AM

Expert Answer

Figure shows the trajectory of charged particle moving in the electric field due to infinite line charge.

Initially the charge particle started with velocity v in the direction that makes angle 30° to the normal drawn to the infinite line charge.

If we resolve the velocity as ( v cos30 ) along the normal and ( v sin30 ) along the perpendiculardirection to normal,

then ( v cos30 ) component is affected due to force F of repulsion due to electric field of infinite line charge.

The vertical componenet ( v sin30 ) is unaffected.

Hence the charged particle move in a parabolic path like projectile motion in gravitational field.

If v_i = ( v cos30 ) is the initial velocity at A along the direction of normal and the velocity becomes zero at B,

then by work-energy theorem ,

begin mathsize 14px style 1 half m space v subscript i superscript 2 space equals space minus integral subscript r subscript i end subscript superscript r subscript f end superscript F space d r end style

...............................(1)

where F is the force due to electric field

begin mathsize 14px style equals space q E space equals space q fraction numerator lambda over denominator 2 pi epsilon subscript o r end fraction end style

.....................(2)

where λ is the charge per unit length in the infinite line charge

hence eqn.(1) is written as,

begin mathsize 14px style 1 half m space v subscript i superscript 2 space equals space fraction numerator q lambda over denominator 2 pi epsilon subscript o end fraction integral subscript r subscript f end subscript superscript r subscript i end superscript fraction numerator d r over denominator r end fraction space equals space fraction numerator q lambda over denominator 2 pi epsilon subscript o end fraction space ln open parentheses r subscript i over r subscript f close parentheses end style

...................(3)

By substituting v_i = ( v cos30 ) in above eqn.(3), we get , r_f = r_i exp[ - ( 3m v² π ε_o ) / ( 4 q λ ) ]

Answered by Thiyagarajan K | 21 Jan, 2020, 11:04: PM

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JEE Class main Answered

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