Two identical particles have the same mass and the same charge. The mass of each particle is 3.346×10^−27 kg. What is the charge on each particle if the ratio of the gravitational force to the electrostatic force is 1.619×10^−36?
5.135×10^−38 C
1.947×10^37 C
2.266×10^−19 C
4.413×10^18 C
Answer is 2.266x10^-19 C
To find the charge on each particle, we can equate the expressions for gravitational force and electrostatic force.
The gravitational force between the particles can be calculated using Newton's law of gravitation:
F_grav = G * (m1 * m2) / r^2
Where:
- F_grav is the gravitational force,
- G is the gravitational constant,
- m1 and m2 are the masses of the particles, and
- r is the distance between the particles.
The electrostatic force between the particles can be calculated using Coulomb's law:
F_electro = k * (q1 * q2) / r^2
Where:
- F_electro is the electrostatic force,
- k is the electrostatic constant,
- q1 and q2 are the charges on the particles, and
- r is the distance between the particles.
Given that the ratio of gravitational force to electrostatic force is 1.619×10^−36, we can set up the equation:
F_grav / F_electro = 1.619×10^−36
Plugging in the formulas for the gravitational and electrostatic forces, we get:
(G * (m1 * m2) / r^2) / (k * (q1 * q2) / r^2) = 1.619×10^−36
Simplifying, we find:
G * (m1 * m2) = k * (q1 * q2) * 1.619×10^−36
We are given that the masses of the particles are the same and equal to 3.346×10^−27 kg. So, we can substitute this value in:
G * ((3.346×10^−27 kg) * (3.346×10^−27 kg)) = k * (q1 * q2) * 1.619×10^−36
Simplifying, we have:
G * (3.346×10^−27 kg)^2 = k * (q1 * q2) * 1.619×10^−36
Now, let's calculate the value of the left-hand side of the equation:
G * (3.346×10^−27 kg)^2 = (6.67430×10^−11 N(m^2/kg^2)) * [(3.346×10^−27 kg)^2] = 2.38916826×10^−47 N(m^2/kg^2)
Therefore, our equation becomes:
2.38916826×10^−47 N(m^2/kg^2) = k * (q1 * q2) * 1.619×10^−36
Rearranging the equation to solve for the product of the charges:
q1 * q2 = (2.38916826×10^−47 N(m^2/kg^2)) / (k * 1.619×10^−36)
To find the charge on each particle, we take the square root of this product:
q1 = sqrt((2.38916826×10^−47 N(m^2/kg^2)) / (k * 1.619×10^−36))
Using the appropriate values for the constants:
q1 = sqrt((2.38916826×10^−47 N(m^2/kg^2)) / ((8.9875×10^9 N(m^2/C^2)) * (1.619×10^−36)))
Simplifying the equation:
q1 = sqrt(2.38916826×10^−12 C)
q1 ≈ 4.886×10^−7 C
Since the particles are identical, the charge on each particle is the same. Therefore, the charge on each particle is approximately 4.886×10^−7 C.
None of the given answer choices match this result. Thus, none of the given options are correct.
To find the charge on each particle, we need to compare the gravitational force to the electrostatic force. The formula for the gravitational force between two particles is given by:
F_gravity = (G * m^2) / r^2
where G is the gravitational constant, m is the mass of each particle, and r is the distance between the particles. The formula for the electrostatic force between two particles is given by:
F_electrostatic = (k * q^2) / r^2
where k is the electrostatic constant, q is the charge on each particle, and r is the distance between the particles.
Since the two particles are identical, their mass (m) and charge (q) are the same. Therefore, we can simplify our comparison of the two forces by dividing the gravitational force equation by the electrostatic force equation:
(F_gravity / F_electrostatic) = (G * m^2) / (k * q^2)
Given that the ratio of the gravitational force to the electrostatic force is 1.619×10^−36, we can substitute this value into the equation:
1.619×10^−36 = (G * m^2) / (k * q^2)
Now we can solve for the charge on each particle (q):
q^2 = (G * m^2) / (k * 1.619×10^−36)
We need to find q, so we take the square root of both sides:
q = sqrt((G * m^2) / (k * 1.619×10^−36))
Now we can substitute the given values into the equation:
q = sqrt((6.67430×10^−11 m^3 kg^−1 s^−2 * (3.346×10^−27 kg)^2) / (8.988×10^9 N m^2 C^−2 * 1.619×10^−36))
Simplifying the expression within the square root:
q = sqrt(((6.67430×10^−11 * (3.346×10^−27)^2) / (8.988×10^9 * 1.619×10^−36)) kg^3 m^3 / (N C^−2 s^2))
Note that the units cancel out, leaving us with just the charge in Coulombs (C). Evaluating the expression:
q = sqrt(7.40419159×10^−82 C^2)
Taking the square root:
q = 2.72088×10^−41 C
Therefore, the charge on each particle is approximately 2.72088×10^−41 C, which is not one of the options provided.
Coulomb force is Fc = kq^2/d^2
So, q^2 = Fcd^2/k
Gravity force is Fg=Gm^2/d^2
so, d^2 = Gm^2/Fg
Thus,
q^2 = Fc/Fg * Gm^2/k
You have Fc/Fg and m^2, so get G and k and crank it out