Two protons in a molecule are 3.80 multiplied by 10-10 m apart. Find the electrical force exerted by one proton on the other.
Magnitude
N
(b) State how the magnitude of this force compares with the magnitude of the gravitational force exerted by one proton on the other.
(electrical force / gravitational force)
(c) What if? What must be a particle's charge-to-mass ratio if the magnitude of the gravitational force between two of these particles is equal to the magnitude of electrical force between them?
C/kg
To find the electrical force exerted by one proton on the other, we can use Coulomb's Law. Coulomb's Law states that the electrical force between two charged particles is directly proportional to the product of their charges and inversely proportional to the square of the distance between them.
Let's break down the problem step by step:
(a) To find the electrical force exerted by one proton on the other, we need to know the charges of the protons. Protons have an elementary charge of +1.6 x 10^-19 C.
Let q₁ be the charge of the first proton and q₂ be the charge of the second proton. Since both protons have the same charge, we have q₁ = q₂ = +1.6 x 10^-19 C.
The distance between the protons is given as 3.80 x 10^-10 m.
Now we can plug in these values into the Coulomb's Law equation:
Force = (k * |q₁ * q₂|) / r²
Where k is the Coulomb's constant, approximately equal to 9 x 10^9 N m²/C², |q₁ * q₂| is the absolute value of the product of the charges, and r is the distance between the charges.
Let's calculate the electrical force:
Force = (9 x 10^9 N m²/C² * |1.6 x 10^-19 C * 1.6 x 10^-19 C|) / (3.80 x 10^-10 m)²
Force ≈ 2.3 x 10^-8 N (rounded to two significant figures)
Therefore, the magnitude of the electrical force exerted by one proton on the other is approximately 2.3 x 10^-8 N.
(b) To compare the magnitude of the electrical force with the magnitude of the gravitational force exerted by one proton on the other, we need to know the gravitational force equation.
The gravitational force between two particles can be calculated using Newton's Law of Universal Gravitation:
Force_gravity = (G * |m₁ * m₂|) / r²
Where G is the gravitational constant, which is approximately equal to 6.674 x 10^-11 N m²/kg², |m₁ * m₂| is the absolute value of the product of the masses, and r is the distance between the masses.
Since we're dealing with protons, we can assume their masses to be equal. Let m be the mass of each proton.
By comparing the two forces, we get:
(electrical force / gravitational force) = [(k * |q₁ * q₂|) / r²] / [(G * |m₁ * m₂|) / r²]
Since the distances r² and masses m₁ and m₂ are the same, they cancel out, leaving us with:
(electrical force / gravitational force) = (k * |q₁ * q₂|) / (G * |m * m|)
Plugging in the values:
(electrical force / gravitational force) = [(9 x 10^9 N m²/C²) * |1.6 x 10^-19 C * 1.6 x 10^-19 C|] / [(6.674 x 10^-11 N m²/kg²) * |1.67 x 10^-27 kg * 1.67 x 10^-27 kg|]
(electrical force / gravitational force) ≈ 3.6 x 10^36
Therefore, the magnitude of the electrical force is approximately 3.6 x 10^36 times larger than the magnitude of the gravitational force exerted by one proton on the other.
(c) If the magnitude of the gravitational force between two particles is equal to the magnitude of the electrical force between them, we have the equation:
|G * |m * m||| = |k * |q₁ * q₂|||
Simplifying:
|G * |m * m||| / |k * |q₁ * q₂||| = 1
We need to find the charge-to-mass ratio, q/m, for this equality.
Rewriting the equation in terms of q/m:
(q/m) = (|G * |m * m||| / |k * |q₁ * q₂|||)^(1/2)
Plugging in the values for G, k, m, q₁, and q₂:
(q/m) = [(6.674 x 10^-11 N m²/kg² * |1.67 x 10^-27 kg * 1.67 x 10^-27 kg|) / (9 x 10^9 N m²/C² * |1.6 x 10^-19 C * 1.6 x 10^-19 C|)]^(1/2)
(q/m) ≈ 5.9 x 10^7 C/kg (rounded to two significant figures)
Therefore, if the magnitude of the gravitational force between two particles is equal to the magnitude of the electrical force between them, the charged particle's charge-to-mass ratio must be approximately 5.9 x 10^7 C/kg.