The region between two concentric spheres of radii ‘a’ and ‘b’, respectively (see figure), has volume charge density ρ = A/r , where A is a constant and r is the distance from the centre. At the centre of the spheres is a point charge Q. The value of A such that the electric field in the region between the spheres will be constant is: | Electric Charges and Fields

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 Multiple Choice QuestionsMultiple Choice Questions

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1.

The region between two concentric spheres of radii ‘a’ and ‘b’, respectively (see figure), has volume charge density ρ = A/r , where A is a constant and r is the distance from the centre. At the centre of the spheres is a point charge Q. The value of A such that the electric field in the region between the spheres will be constant is:

  • fraction numerator straight Q over denominator 2 πa squared end fraction
  • fraction numerator straight Q over denominator 2 straight pi left parenthesis straight b to the power of 2 minus end exponent straight a squared right parenthesis end fraction
  • fraction numerator 2 straight Q over denominator straight pi left parenthesis straight a squared minus straight b squared right parenthesis end fraction
  • fraction numerator 2 straight Q over denominator straight pi left parenthesis straight a squared minus straight b squared right parenthesis end fraction


A.

fraction numerator straight Q over denominator 2 πa squared end fraction

A Gaussian surface at distance r from centre.




fraction numerator straight Q space plus integral subscript straight a superscript straight r begin display style straight A over straight r end style 4 straight pi squared dr over denominator straight epsilon subscript straight o end fraction space equals space straight E 4 πr squared
straight E 4 straight epsilon subscript straight o straight r squared space equals space straight Q space plus space straight A fraction numerator 4 straight pi over denominator straight r squared end fraction open parentheses fraction numerator straight r squared minus straight a squared over denominator 2 end fraction close parentheses
straight E space equals fraction numerator 1 over denominator 4 πε subscript straight o end fraction space open square brackets straight Q over straight r squared plus straight A space 2 straight pi open parentheses fraction numerator straight r squared minus straight a squared over denominator straight r squared end fraction close parentheses close square brackets
straight E space equals space fraction numerator 1 over denominator 4 πε subscript straight o end fraction open square brackets straight Q over straight r squared plus straight A 2 straight pi minus fraction numerator straight A 2 πa squared over denominator straight r squared end fraction close square brackets
straight E space equals space fraction numerator 1 over denominator 4 πε subscript straight o end fraction space straight x space straight A space straight x space 2 straight pi
At the centre of the sphere is a point charge Q. The value of A such that the electric field in the region between the spheres will be constant is 
As, Q = 2πAa2
i.e A = Q/2πa2

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2.

A long cylindrical shell carries positive surface charge  in the upper half and negative surface charge  in the lower half. The electric field lines around the cylinder will look like figure given in: (figures are schematic and not drawn to scale)

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3.

Two charges, each equal to q, are kept at x = −a and x = a on the x-axis. A particle of mass m and charge qo =-q/2 is placed at the origin. If charge qo is given a small displacement (y<< a) along the y-axis, the net force acting on the particle is proportional to

  • y

  • -y

  • 1/y

  • 1/y

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4.

A charge Q is uniformly distributed over a long rod AB of length L as shown in the figure. The electric potential at the point O lying at a distance L from the end A is

  • fraction numerator straight Q over denominator 8 πε subscript straight o straight L end fraction
  • fraction numerator 3 space straight Q over denominator 4 space πε subscript straight o straight L end fraction
  • fraction numerator straight Q over denominator 4 πε subscript straight o straight L space In space 2 end fraction
  • fraction numerator straight Q over denominator 4 πε subscript straight o straight L space In space 2 end fraction
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5.

This question has statement 1 and statement 2. Of the four choices given after the statements, choose the one that best describes the two statements.
An insulating solid sphere of radius R has a uniformly positive charge density ρ. As a result of this uniform charge distribution, there is a finite value of the electric potential at the centre of the sphere, at the
surface of the sphere and also at a point out side the sphere. The electric potential at infinity is zero.

Statement 1: When a charge q is taken from the centre to the surface of the sphere, its potential energy changes by qρ/3εo
Statement 2: The electric field at a distance r(r < R) from the centre of the sphere is  ρr/3εo

  • Statement 1 is true, Statement 2 is true, Statement 2 is not the correct explanation for statement 1.

  • Statement 1 is true, Statement 2 is false

  • Statement 1 is false, Statement 2 is true

  • Statement 1 is false, Statement 2 is true

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6.

In a uniformly charged sphere of total charge Q and radius R, the electric field E is plotted as a function of distance from the centre. The graph which would correspond to the above will be

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7.

Let there be a spherically symmetric charge distribution with charge density varying as  space straight rho left parenthesis straight r right parenthesis space equals space space straight rho subscript 0 space open parentheses 5 over 4 minus straight r over straight R close parentheses upto r = R, and ρ(r) = 0 for r > R, where r is the distance from the origin. The electric field at a distance r ( r < R) from the origin is given by

  • fraction numerator 4 space straight pi space straight rho subscript 0 straight r over denominator 3 straight epsilon subscript 0 end fraction space open parentheses 5 over 3 minus straight r over straight R close parentheses
  • fraction numerator begin display style space straight rho subscript 0 straight r end style over denominator begin display style 4 straight epsilon subscript 0 end style end fraction space open parentheses fraction numerator begin display style 5 end style over denominator begin display style 3 end style end fraction minus fraction numerator begin display style straight r end style over denominator begin display style straight R end style end fraction close parentheses
  • fraction numerator 4 space straight rho subscript 0 straight r over denominator 3 straight epsilon subscript 0 end fraction space open parentheses 5 over 3 minus straight r over straight R close parentheses
  • fraction numerator 4 space straight rho subscript 0 straight r over denominator 3 straight epsilon subscript 0 end fraction space open parentheses 5 over 3 minus straight r over straight R close parentheses
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8.

A thin spherical shell of radius R has charge Q spread uniformly over its surface. Which of the following graphs most closely represents the electric field E(r) produced by the shell in the range 0 ≤ r< ∞ , where r is the distance from the centre of the shell? 

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9.

Charges are placed on the vertices of a square as shown. Let E be the electric field and V the potential at the centre. If the charges on A and B are interchanged with those on D and C respectively, then

  • straight E with rightwards arrow on top remains unchanged, V changes 
  • Both straight E with rightwards arrow on top and V change

  • straight E with rightwards arrow on top and V remain unchanged
  • straight E with rightwards arrow on top and V remain unchanged
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10.

A charged particle with charge q enters a region of constant, uniform and mutually orthogonal fields straight E with rightwards arrow on top space and space straight B with rightwards arrow on top, with a velocity straight v with rightwards arrow on top perpendicular to both straight E with rightwards arrow on top space and space straight B with rightwards arrow on top , and comes out without any change in magnitude or direction of straight v with rightwards arrow on top .Then

  • straight v with rightwards arrow on top space equals space straight E with rightwards arrow on top space straight x space straight B with rightwards arrow on top space divided by straight B squared
  • straight v with rightwards arrow on top space equals space straight B with rightwards arrow on top space straight x space straight E with rightwards arrow on top divided by straight B squared
  • straight v with rightwards arrow on top space equals space straight E with rightwards arrow on top space straight x space straight B with rightwards arrow on top divided by straight E squared
  • straight v with rightwards arrow on top space equals space straight E with rightwards arrow on top space straight x space straight B with rightwards arrow on top divided by straight E squared
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