How many times is the electromagnetic force stronger then Gravitational force (a) \(10^{13}\) (b) \(10^{-13}\) (c) \(10^{36}\) (d) \(10^{-36}\)

To compare the electromagnetic force and the gravitational force, we need to know their formulas. The gravitational force between two masses (m1 and m2) is given by: \(F_g = G\dfrac{m_1 m_2}{r^2}\) where G is the gravitational constant, and r is the distance between the masses. The electromagnetic force between two charges (q1 and q2) is given by: \(F_e = k \dfrac{q_1 q_2}{r^2}\) where k is the electrostatic constant, and r is the distance between the charges.

Now, we will find the ratio of the electromagnetic force to the gravitational force: \(\dfrac{F_e}{F_g} = \dfrac{k \dfrac{q_1 q_2}{r^2}}{G \dfrac{m_1 m_2}{r^2}}\) The term r^2 cancels out in the numerator and the denominator: \(\dfrac{F_e}{F_g} = \dfrac{k q_1 q_2}{G m_1 m_2}\)

Let's take the proton and the electron, as these particles have both mass and charge so that the forces do not become zero. The mass of the proton is \(m_p = 1.67 \times 10^{-27} kg\) and the mass of the electron is \(m_e = 9.11 \times 10^{-31} kg\). The charge of the proton is \(q_p = 1.60 \times 10^{-19} C\) and the charge of the electron is \(q_e = -1.60 \times 10^{-19} C\). Note that the charges are equal in magnitude but opposite in sign. We also know \(G = 6.674 \times 10^{-11} N m^2 kg^{-2}\) and \(k = 8.99 \times 10^9 N m^2 C^{-2}\). We can now plug in these values into the ratio formula: \(\dfrac{F_e}{F_g} = \dfrac{(8.99 \times 10^9) (1.60 \times 10^{-19})^2}{(6.674 \times 10^{-11}) (1.67 \times 10^{-27}) (9.11 \times 10^{-31})}\) After calculating this expression, we get: \(\dfrac{F_e}{F_g} \approx 2.26 \times 10^{36}\)

Now that we've found that the electromagnetic force is approximately \(2.26 \times 10^{36}\) times stronger than the gravitational force, we can choose the correct option: Since \(2.26 \times 10^{36}\) is very close to \(10^{36}\), we can choose the option (c) \(10^{36}\).