Question

1. Two identical solid copper spheres of radius r placed in contact with each other. The gravitational attraction is proportional to : ?

2. Why 1.6 multiply 10 to the power -20 coulombs charge is not possible, where as 1.6 multiply 10 to the power -18 coulombs is possible

3. Clock a based on spring oscillations and a clock b based on oscillations of simple pendulum are synchronized on Earth. Both are taken to Mars. Where mass is 0.1 times mass of Earth and Radius is half of Earth's radius. What are the measures of time in both clocks now.

4. A soap solution has bubbles in it. The surface tension of that is S. Radius of a bubble is R and the pressure inside the bubble is atmospheric pressure. What is the electric potential on the surface of bubble?

5. An electric field of 30 n coulomb exists along the negative x axis in space. Calculate the potential difference Vb - Va where the points a and b are given by a=(0,0)and b(0,2)

6. If three infinite charged sheets of uniform surface charge densities of sigma ,2 sigma, - 4 sigma respectively are placed parallel to each other, Find electric field at different points.

7. Two conductors are of uniform shape and size one of them is made of copper, while the other is made of aluminum (less conductive). The charge induced in aluminum will be equal to copper or less or more ?

Answer 1

1.   F = G M * M / x^2 distance x = 2 R = d = diameter M =  (density p) * 4 pi /3  R^3      = 4 p * pi /3 * d^3 /8    F = G M^2 / x^2    = G (p^2 pi^2/6^2) d^6 /d^2   So F is  proportional to d^4 Mass is dep. on diameter as density of the material is  a constant and  Distance = diameter. This happens only because they are in contact... not in general...

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2.  Electron charge e  is the smallest charge quantum possible.  This is established by quantum physics.  Charge exists in integral  multiples of  e.  So  q = 1.6 * 10^-20 coul  is not possible. q = 1.6 * 10^-18  = 10 e  ie., there are either 10 additional electrons or 10 electrons are taken away.

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3.    Time period T1 = 2 pi * root (m/k) for spring based clock a.
       time period T2  =  2 pi * root(L/g)  for clock b  based on pendulum.

      g on Mars / g on Earth = ( M2 / R2^2 ) / (M1 /R1^2)
        g'  = 2 g

     So  T1' = T1   remains same on Mars and Earth.
      T2' = T2/root(2)
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4.  Pressure difference between outside and inside of bubble
         = P2 - P1 = 4 S / R   

Force exerted by this pressure difference on a small area on surface A
       =  F = 4 S / R * A  

Let there be an electric charge on the soap film.  Charge density = sigma     

Electric field E at the surface =  sigma / (2 * epsilon_0)     
Electric Force on area A =  E * charge = sigma * (sigma * A) / (2 * epsilon).
 Electric force tries to explode the bubble.

 Surface tension tries to contract the bubble.
    if the bubble is in equilibrium =>   sigma^2 = 8 S epsilon / R

    Electric potential V at the surface of bubble = q / (4 pi epsilon * R)
    q = 4 pi R^2 * sigma
     V =  sigma  * R / epsilon = sqroot[8 S R/ epsilon ]
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 5.  Electric field E is directed in the negative X direction.  E points from b towards a.
         so E = - E i
        Vb - Va =  E * distance = 2 E
    Potential drops along the direction of E.  
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6.    Sheets are S1, S2 and S3.  S3 is negatively charged.  The field is directed away from the sheet for S1 and S2.   Field due to S3 is directed towards S3. 

The magnitudes of the electric fields due to them respectively are:
            E1 = sigma / (2 epsilon) = E  ,  say
            E2 =  2 sigma / ( 2 epsilon) = 2 E
            E3  =  4 sigma / (2 epsilon) = 4 E

  Take a point A  on the left of all sheets.   Point B between S1 and S2.  Point C between S2 and S3.  and point D is to the right of S3.

    net field E at A =  - E1 - E2  + E3 = E    towards S3.   
    net field at B =  E1 - E2 + E3 = 3 E towards S3   
    net field at C =  E1 + E2 + E3 = 7 E   
     field at D =  E1 + E2 - E3 = - E   towards S3.

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7.   Copper and Aluminum are good conductors.  The Aluminum has more resistance than copper.  But both conduct electricity well enough..  When a charge is brought near them, then equal and opposite charge is induced in either case...  There is perhaps a little more energy is spent to induce charges in Aluminum.  Electrons get either attracted or repelled towards the inducing charge.  Metals allow flow of electrons , so this property is needed.   Resistance causes heat generation.  All metals will have equal and opposite charge induced on them  during induction.