Question

Proton is placed 100 micrometers from a helium nucleus. Gravity pulls the proton and nucleus together while the electric force pushes them apart. Which is stronger and by how much ?

Answer 1

Electric force is
`6.2\cdot 10^(35)` times stronger than gravitational force

Step-by-step explanation:

The gravitational force between two objects is given by:

`F_G = G(m_1 m_2)/(r^2)`

where

G is the gravitational constant

m1, m2 are the masses of the two objects

r is the separation between the objects

Here we have:

`m_1 = 1.67\cdot 10^(-27)kg` (mass of the proton)

`m_2=4\cdot 1.67\cdot 10^(-27) =6.68\cdot 10^(-27) kg` (mass of the helium nucleus is equal to 4 times the mass of a proton)

`r=100 \mu m = 100\cdot 10^(-6) m`

So,

`F_G = (6.67\cdot 10^(-11)) ((1.67\cdot 10^(-27))(6.68\cdot 10^(-27)))/((100\cdot 10^(-6))^2)=7.44\cdot 10^(-56) N`

The electric force between two charged object is given by

`F_E=k(q_1 q_2)/(r^2)`

where

k is the Coulomb constant

q1, q2 are the two charges

r is the separation

Here we have

`q_1 = 1.6\cdot 10^(-19)C` (charge of the proton)

`q_2 = 2\cdot (1.6\cdot 10^(-19))=3.2\cdot 10^(-19)C` (charge of the helium nucleus is twice that of the proton)

`r=100 \mu m = 100\cdot 10^(-6) m`

So,

`F_E=(9\cdot 10^9) ((1.6\cdot 10^(-19))(3.2\cdot 10^(-19)))/((100\cdot 10^(-6))^2)=4.6\cdot 10^(-20)N`

Therefore, we see that the electric force is much stronger than the gravitational force, by a factor of:

`(F_E)/(F_G)=(4.6\cdot 10^(-20))/(7.44\cdot 10^(-56))=6.2\cdot 10^(35)`