Numerical-2

Q. Coulomb’s law for electrostatic force between two point charges and Newton’s law for gravitational force between two stationary point masses, both have inverse-square dependence on the distance between the charges/masses. (a) Compare the strength of these forces by determining the ratio of their magnitudes (i) for an electron and a proton and (ii) for two protons. (b) Estimate the accelerations of electron and proton due to the electrical force of their mutual attraction when they are 1 Å (= 10^-10 m) apart? (mp = 1.67 ×10^–27 kg, me = 9.11 × 10^–31 kg).

Solution :

(a) (i) The electric force between an electron and a proton at a distance r apart is:

F(e) = -e^2 / 4πεo r^2

where the negative sign indicates that the force is attractive. The

corresponding gravitational force (always attractive) is:

F(G) = - G mp me / r^2

where mp and me are the masses of a proton and an electron

respectively.

| F(e) / F(G) | = [-e^2 / 4πεo r^2]/ [- G mp me / r^2]= 2.4 x 10 ^39

(ii) On similar lines, the ratio of the magnitudes of electric force

to the gravitational force between two protons at a distance r

apart is : 

| F(e) / F(G) | =[-e^2 / 4πεo r^2]/ [- G mp mp / r^2]=1.3 x 10^36

However, it may be mentioned here that the signs of the two forces

are different. For two protons, the gravitational force is attractive

in nature and the Coulomb force is repulsive . The actual values

of these forces between two protons inside a nucleus (distance

between two protons is ~ 10^-15 m inside a nucleus) are F(e) ~ 230 N whereas F(G)  ~ 1.9 × 10^–34 N.

The (dimensionless) ratio of the two forces shows that electrical

forces are enormously stronger than the gravitational forces.

(b) The electric force F exerted by a proton on an electron is same in

magnitude to the force exerted by an electron on a proton; however

the masses of an electron and a proton are different. Thus, the

magnitude of force is 

| F | = e^2 / 4πεor^2 =8.987 × 109 Nm2 /C2 × (1.6 ×10^–19C)^2 / (10–^10m)^2

| F |= 2.3 × 10^–8 N 

Using Newton’s second law of motion, F = ma, the acceleration

that an electron will undergo is a = 2.3×10^–8 N / 9.11 ×10^–31 kg = 2.5 × 10^22 m/s2

Comparing this with the value of acceleration due to gravity, we

can conclude that the effect of gravitational field is negligible on

the motion of electron and it undergoes very large accelerations

under the action of Coulomb force due to a proton.

The value for acceleration of the proton is

2.3 × 10^–8 N / 1.67 × 10^–27 kg = 1.4 × 10^19 m/s2