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Question : 31
Total: 35
(a) (i) Use Gauss' 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 densityλ v λ
(iii) Draw a graph of kinetic energy as a function of linear charge densityλ
OR
(b) (i) Consider two identical point charges located at points( 0 , 0 ) ( a , 0 )
(1) Is there a point on the line joining them at which the electric field is zero?
(2) Is there a point on the line joining them at which the electric potential is zero?
Justify your answers for each case.
(ii) State the significance of negative value of electrostatic potential energy of a system of charges.
Three charges are placed at the corners of an equilateral triangleABC 2.0 m
(ii) An infinitely long positively charged straight wire has a linear charge density
(iii) Draw a graph of kinetic energy as a function of linear charge density
OR
(b) (i) Consider two identical point charges located at points
(1) Is there a point on the line joining them at which the electric field is zero?
(2) Is there a point on the line joining them at which the electric potential is zero?
Justify your answers for each case.
(ii) State the significance of negative value of electrostatic potential energy of a system of charges.
Three charges are placed at the corners of an equilateral triangle
Solution: 👈: Video Solution
(a) (i) Electric field due to an infinitely long thin straight wire with uniform linear charge density:
Linear charge density= λ
A pointP x
A cylindrical Gaussian surface of lengthl x
Magnitude of electric field at every point the curved surface of the cylinder is same and is directed outward making an angle0 ∘
Electric fields at the two flat surfaces of the cylinder are zero since, the direction of electric field through those surfaces and the direction of area vector are perpendicular.
So, total flux through the curved surface
= ϕ = E × 2 π x l
According to Gauss' law,ϕ =
∴ E × 2 π x l =
∴ E =
(ii) Electric field produced by an infinitely long straight charged wire,
E =
This field provides the centripetal force to the revolving electron.
e E =
or,
=
or,m v 2 =
∴ ( KE ) =
m v 2 =
(iii) The required equation,
m v 2 =
It is the equation of a straight line passing through the origin.
OR
(b) (i) (1) Yes, Electric field will be zero at(
, 0 )
(2) Electric potential will not be zero at any point since at every point the potential will be the summation of both the potentials and both charges are of similar nature.
(ii) Significance of negative electrostatic potential energy:
Negative electrostatic potential energy of a system of charges means work is to be done against the field to move the charges apart.
Potential energy due to charges at A and B
U 1 =
Or, U 1 =
∴ U 1 = − 72 × 10 − 3 J
Potential energy due to charges atA C
U 2 =
Or, U 1 =
∴ U 2 = 36 × 10 − 3 J
Potential energy due to charges atB C
U 3 =
Or, U 3 =
∴ U 3 = − 36 × 10 − 3 J
Net potential energy of the system of three charges.
U = U 1 + U 2 + U 3
Or, U = − 72 × 10 − 3 J + 36 × 10 − 3 J − 36 × 10 − 3 J
∴ U = − 72 × 10 − 3 J
Linear charge density
A point
A cylindrical Gaussian surface of length
Magnitude of electric field at every point the curved surface of the cylinder is same and is directed outward making an angle
Electric fields at the two flat surfaces of the cylinder are zero since, the direction of electric field through those surfaces and the direction of area vector are perpendicular.
So, total flux through the curved surface
According to Gauss' law,
(ii) Electric field produced by an infinitely long straight charged wire,
This field provides the centripetal force to the revolving electron.
or,
or,
(iii) The required equation,
It is the equation of a straight line passing through the origin.
OR
(b) (i) (1) Yes, Electric field will be zero at
(2) Electric potential will not be zero at any point since at every point the potential will be the summation of both the potentials and both charges are of similar nature.
(ii) Significance of negative electrostatic potential energy:
Negative electrostatic potential energy of a system of charges means work is to be done against the field to move the charges apart.
Potential energy due to charges at A and B
Potential energy due to charges at
Potential energy due to charges at
Net potential energy of the system of three charges.
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