Question

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Answer 1

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Question 1 :

Given, total charge on the rod (Q) = 8.30 nC

length of the rod (L) = 1.40 cm

distance of point P from the center of the rod (l) =8.80 cm = 8.80 x 10^-2 m

So, charge per unit length on the rod () is

I have taken a charge element dx at a distance x from the origin O on the rod.

Distance of the charge element from the point P is , r = ( l + L/2 - x ) .

Charge on the element dx =

So, electric field due to this element at point P is

  

Where,

    is constant .

and is also a constant .

So,    ..............................(1)

NOTE :

This type of integration can be done as follow :

  

Now from equation (1), we get

  

  

putting all the values, we get

E=9×109 × 5.93 × 10-7 × 1.40 x 10-2 (8.8 × 10-2)2-( 1.40×10-2 ) 2

  

74.7 76.95 × 10-1 9.7 × 10-M/C

And the direction of this field will always along the positive X-axis.

Question 2 :

This is simply a case of projectile motion . Here the acting force is electrostatic instead of gravitational.

We know the equation for range (R) in projectile motion (in case of gravitation) :

here, g is the acceleration due to gravity which always acts downward.

u is the initial velocity of the projectile and

is the launch angle of projectile with x - axis in anticlockwise direction

Comparing this situation with our problem.

Here, our projectile is electron. The positive plate of capacitor will always attract the electron towards it .

Consider the positive plate as x- axis and a line from positive plate to negative plate as y -axis.

So, instead of g (acceleration due to gravity) here the acceleration of the electron will be

(from Newton's second law of motion )

Where, F_e is electrostatic force and  

F_e = eE (E is electric field)

So,

So, we can write equation for range in this case as

Given,

R = 2.50 cm = 2.50 x 10^-2 m

u = 2.45 x 10⁶ m/s

θ = 45°

Putting all the values in equation for R, we get

or,

or,

We have,

Where, e = 1.602 x 10^-19 C is electronic charge

m_e = 9.1 x 10^-31 kg is mass of electron

So,

or,

Or,

This is strength of electric field between the plates of the capacitor.

For any doubt please comment and please give an up vote. Thank you.