Define the term electric field intensity. Write its SI unit. Derive an expression for the electric field intensity at a point on the axis of an electric dipole. 

The force experienced by a unit positive charge placed at a point is termed as the electric field intensity at that point. It is vector quantity it’s direction is in the force acting on +ive charge.
The SI unit of electric field intensity is NC–1. Electric field at an axial point of electric dipole:
Assume point P is located at distance r from the centre of an electric dipole as shown in the figure.

The force experienced by a unit positive charge placed at a point is

The force experienced by a unit positive charge placed at a point is

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Using Gauss's law, derive an expression for the electric field intensity at any point outside a uniformly charged thin spherical shell of radius R and charge density a C/m2. Draw the field lines when the charge density of the sphere is (i) positive, (ii) negative. (b) A uniformly charged conducting sphere of 2.5 m in diameter has a surface charge density of 100 μC/m2. Calculate the
(i) charge on the sphere
(ii) total electric flux passing through the sphere.

(a) Electric field intensity at any point outside a uniformly charged spherical shell:

(a) Electric field intensity at any point outside a uniformly charged
Consider a thin spherical shell of radius R with centre O. Let charge +q is uniformly distributed over the surface of the shell.
Let P be any point on the sphere S1 with centre O and radius r. According to Gauss's Law

(a) Electric field intensity at any point outside a uniformly charged

If σ is charge density,
∴   straight q equals 4 πr squared
∴   Electric field lines when the charged density of the sphere:
(i) Positive            (ii)     Negative

(a) Electric field intensity at any point outside a uniformly charged
(b) Here diameter = 2.5 m                therefore space space straight R space equals space 25 over 2 space equals space 12.5 space straight m
Charge density   straight sigma space equals space 100 space μc divided by straight m squared space equals space 100 space cross times space 10 to the power of negative 6 end exponent space equals space 10 to the power of negative 4 end exponent space straight c divided by straight m squared
  (i)                      straight q space equals space 4 straight pi space straight R squared straight sigma
                                space equals space 4 space cross times space 3.14 space cross times space left parenthesis 1.25 right parenthesis squared space cross times space 10 to the power of negative 4 end exponent
space equals space 19.625 space cross times space 10 to the power of negative 4 end exponent
equals space 1.96 space cross times space 10 to the power of negative 3 end exponent space straight C
(ii) Total electric flux
                     straight ϕ subscript straight E space equals space straight q over element of subscript 0
therefore             straight ϕ subscript straight E space equals space fraction numerator 1.96 space cross times space 10 to the power of negative 13 end exponent over denominator 8.85 space cross times space 10 to the power of negative 12 end exponent end fraction space equals space 0.221 space cross times space 10 to the power of 9 space equals space 2.21 space cross times space 10 to the power of 8 space Nm squared space straight C to the power of negative 1 end exponent

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Two fixed point charges + 4e and + e units are separated by a distance a. Where should the third point charge be placed for it to be in equilibrium?


Let a point charge q be held at a distance x from the charge + 4e, figure given below

Let a point charge q be held at a distance x from the charge + 4e, fi

∴ Distance of q from charge + e = (a – x) Force on this charge exerted by the charge + 4e is
straight F subscript 1 space equals space fraction numerator straight q space left parenthesis 4 straight e right parenthesis over denominator 4 straight pi space element of subscript 0 space straight x squared end fraction directed away from (4e)
Force on this charge exerted by the charge + e
straight F subscript 2 space equals space fraction numerator straight q left parenthesis straight e right parenthesis over denominator 4 straight pi space element of subscript 0 left parenthesis straight a minus straight x right parenthesis squared end fraction comma directed away from (e)

For the charge q to be in equilibrium F1 = F2
i.e.,            fraction numerator straight q space left parenthesis 4 straight e right parenthesis over denominator 4 space straight pi space element of subscript 0 space straight x squared end fraction space equals space fraction numerator straight q left parenthesis straight e right parenthesis over denominator 4 space straight pi space element of subscript 0 space left parenthesis straight a minus straight x right parenthesis squared end fraction
or,                   4 over straight x squared space equals space fraction numerator 1 over denominator left parenthesis straight a minus straight x right parenthesis squared end fraction space space space or space space space space 2 over straight x space equals space fraction numerator 1 over denominator straight a minus straight x end fraction
therefore          straight x space equals space 2 straight a minus 2 straight x
or,         3 straight x equals 2 straight a space space space or space space space straight x space equals space 2 straight a divided by 3
Hence the charge q should be held at a distance 2a / 3 from charge (+ 4e).
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Point charges having values + 0.1 μC, + 0.2 μC, – 0.3 μC and –0.2 μC are placed at the corners A, B, C and D respectively of a square of side one metre. Calculate the magnitude of the force on a charge of + 1 μC placed at the centre of the square.

            AC squared space equals space AB squared plus BC squared space equals space 1 plus space 1 space equals space 2
or,                AC space equals space square root of 2 space straight m
           AO space equals space 1 half square root of 2 space straight m space equals space 0.5 space square root of 2 space straight m
Also,   AO space equals space CO space equals space BO space equals space DO space equals space 0.5 square root of 2 space straight m

Let Fa be the force exerted by the charge of ‘+ O.1μC’ at A on the charge of  ‘ + 1 μC’ at the centre O of the square.
Then straight F subscript straight A space equals space 9 space cross times space 10 to the power of 9 space Nm squared straight C to the power of negative 2 end exponent space fraction numerator left parenthesis 0.1 space cross times space 10 to the power of negative 6 end exponent straight C right parenthesis space left parenthesis 1 space cross times space 10 to the power of negative 6 end exponent straight C right parenthesis over denominator 2 left parenthesis 0.5 right parenthesis squared space straight m squared end fraction
or,                         straight F subscript straight A space equals space fraction numerator 9 cross times 10 to the power of 9 cross times 0.1 cross times 10 to the power of negative 12 end exponent over denominator 2 cross times 0.25 end fraction straight N space equals space 0.0018 space straight N

If Fc is the force exerted by charge at C on charge at O, then
                     straight F subscript straight C space equals space 9 space cross times space 10 to the power of 9 space fraction numerator open parentheses 0.3 space cross times space 10 to the power of negative 6 end exponent close parentheses space open parentheses 1 space cross times space 10 to the power of negative 6 end exponent close parentheses over denominator 2 space left parenthesis 0.5 right parenthesis squared end fraction straight N
space space space space space equals space fraction numerator 9 cross times 0.3 cross times 10 to the power of negative 3 end exponent over denominator 2 cross times 0.25 end fraction straight N space equals space 0.0054 space straight N
Both straight F subscript straight A space and space straight F subscript straight C act in the same direction. The resultant of straight F subscript straight A space and space straight F subscript straight C comma
                           straight F subscript 1 space equals space left parenthesis 0.0018 plus 0.0054 right parenthesis thin space straight N space equals space 0.0072 straight N
Force exerted by the charge at B on the charge at O,
                         straight F subscript straight B space equals space 9 cross times 10 to the power of 9 fraction numerator open parentheses 0.2 space cross times space 10 to the power of negative 6 end exponent close parentheses open parentheses 1 space cross times space 10 to the power of negative 6 end exponent close parentheses over denominator 2 left parenthesis 0.5 right parenthesis squared end fraction straight N
space space
                              equals 3.6 space cross times space 10 to the power of negative 3 end exponent straight N space equals space 0.0036 space straight N
Force exerted by the charge at D on the charge at O,
                        straight F subscript straight D space equals space 9 space cross times space 10 to the power of 9 fraction numerator open parentheses 0.2 space cross times space 10 to the power of negative 6 end exponent close parentheses space open parentheses 1 cross times 10 to the power of negative 6 end exponent close parentheses over denominator 2 left parenthesis 0.5 right parenthesis squared end fraction straight N space equals space 0.0036 space straight N
Both straight F subscript straight B space and space straight F subscript straight D act in the same direction. Resultant of straight F subscript straight B space and space straight F subscript straight D apostrophe end subscript
                         straight F subscript 2 space equals space left parenthesis 0.0036 plus 0.0036 right parenthesis space straight N space equals space 0.0072 space straight N

            or,                           Also,
The angle between F1 and F2 is clearly 90°. So, the resultant F of F1 and F2 is given by
straight F space equals space square root of left parenthesis 0.0072 right parenthesis squared plus left parenthesis 0.0072 right parenthesis squared end root straight N space equals space 0.0072 space square root of 2 straight N
space space space equals space 0.0072 space cross times space 1.414 space straight N space equals space 0.01018 space straight N

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(a) Define electric flux. Write its SI units.
(b) The electric field components due to a charge inside the cube of side 0.1 m are as shown:
Ex = α x, where α = 500 N / C–m
Ey = 0, Ez = 0.

Calculate:
(i) the flux through the cube and
(ii) the charge inside the cube.

 


(a) Electric flux:
The total number of electric field lines crossing an area in an electric field is termed as electric flux.
It is denoted by ΦE.
Electric flux is a scalar quantity, its SI unit is Nm2 C–1.
(b) Here, Ex = αx, Ey = 0, Ez = 0
α = 500 N/C–m, side of a cube a = 0.1 m
As the electric field has only X-component then,
straight ϕ subscript straight E space equals space straight E with rightwards arrow on top. space increment straight S with rightwards arrow on top space equals space 0
for each of four faces of cube 1 to Y-axis and Z-axis.
∴ Electric flux is only for left and right face along X-axis of cube.
(i) Electric field at the left face,

(a) Electric flux:The total number of electric field lines crossing a
(ii) According to Gaussian law
                 straight q space equals space straight epsilon subscript 0 straight ϕ
space space space equals space 8.85 space cross times space 10 to the power of negative 12 end exponent space cross times space 0.5
or,              straight q space equals space 4.425 space cross times space 10 to the power of negative 12 end exponent straight C

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