Question 46

A positive ion A and a negative ion B has charges $$6.67\times10^{-19}C$$ and $$9.6\times10^{-10}C$$, and masses $$19.2\times10^{-27}Kg$$ and $$9\times10^{-27}Kg$$ respectively. At an instant, the ions are separated by a certain distance r. At that instant the ratio of the magnitudes of electrostatic force to gravitational force is $$P\times10^{45}$$ , where the value of 10P is (Take $$\frac{1}{4\pi\epsilon_{0}}=9\times10^{9}NM^{2}C^{-1}$$ and universal gravitational constant as $$6.67\times10^{-11}NM^{2}Kg^{-2}$$)
Assume that charge may not be an integral multiple of electrons.


Correct Answer: 5

We need to find the ratio of electrostatic force to gravitational force between the two ions. The electrostatic force (Coulomb's law) is given by $$ F_e = \frac{1}{4\pi\epsilon_0}\frac{q_1 q_2}{r^2} $$. The gravitational force is $$ F_g = \frac{Gm_1 m_2}{r^2} $$. Dividing these gives $$ \frac{F_e}{F_g} = \frac{q_1 q_2}{4\pi\epsilon_0 G m_1 m_2} $$.

The charges are $$q_1 = 6.67 \times 10^{-19}$$ C and $$q_2 = 9.6 \times 10^{-10}$$ C, and the masses are $$m_1 = 19.2 \times 10^{-27}$$ kg and $$m_2 = 9 \times 10^{-27}$$ kg. We also use $$\frac{1}{4\pi\epsilon_0} = 9 \times 10^{9}$$ Nm$$^2$$C$$^{-2}$$ and $$G = 6.67 \times 10^{-11}$$ Nm$$^2$$kg$$^{-2}$$.

Substituting these values into the ratio yields: $$ \frac{F_e}{F_g} = \frac{9 \times 10^{9} \times 6.67 \times 10^{-19} \times 9.6 \times 10^{-10}}{6.67 \times 10^{-11} \times 19.2 \times 10^{-27} \times 9 \times 10^{-27}} $$.

The numerator simplifies as $$ 9 \times 6.67 \times 9.6 \times 10^{9-19-10} = 9 \times 6.67 \times 9.6 \times 10^{-20} = 576.288 \times 10^{-20} = 5.76288 \times 10^{-18} $$. The denominator becomes $$ 6.67 \times 19.2 \times 9 \times 10^{-11-27-27} = 6.67 \times 19.2 \times 9 \times 10^{-65} = 1152.576 \times 10^{-65} = 1.152576 \times 10^{-62} $$.

Therefore, $$ \frac{F_e}{F_g} = \frac{5.76288 \times 10^{-18}}{1.152576 \times 10^{-62}} = 5 \times 10^{44} = 0.5 \times 10^{45} $$. Hence, $$P = 0.5$$ and $$10P = 5$$, giving the final answer as 5.

Create a FREE account and get:

  • Free JEE Mains Previous Papers PDF
  • Take JEE Mains paper tests