(a) 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
(a) Well, it seems like these two protons are getting pretty close to each other! Don't they know about personal space? Anyway, let's calculate the electrical force they're exerting on each other.
We can use Coulomb's Law, which states that the electrical force between two charged particles is given by the equation F = k * (q1 * q2 / r^2), where F is the force, k is the electrostatic constant, q1 and q2 are the charges of the particles, and r is the distance between them.
Since both protons have the same charge (positive) and magnitude (1.6 x 10^-19 C), we can simplify the equation further. Plugging in the values, we get:
F = (8.99 x 10^9 N m^2/C^2) * ((1.6 x 10^-19 C)^2 / (3.8 x 10^-10 m)^2)
After crunching the numbers, we find that the magnitude of the electrical force between the protons is approximately 3.63 x 10^-8 N.
(b) Now let's compare the strength of the electrical force with the gravitational force between the protons. The gravitational force can be calculated using Newton's Law of Universal Gravitation, which states that the force is given by F = G * (m1 * m2 / r^2), where F is the force, G is the gravitational constant, m1 and m2 are the masses of the particles, and r is the distance between them.
The masses of protons are approximately 1.67 x 10^-27 kg, and the distance between them is the same as before (3.8 x 10^-10 m). Plugging in these values, we get:
F_grav = (6.67 x 10^-11 N m^2/kg^2) * ((1.67 x 10^-27 kg)^2 / (3.8 x 10^-10 m)^2)
After doing some more number crunching, we find that the magnitude of the gravitational force between the protons is approximately 9.1 x 10^-52 N.
So, to answer your question, the magnitude of the electrical force (3.63 x 10^-8 N) is much stronger than the magnitude of the gravitational force (9.1 x 10^-52 N). It's like comparing Hulk's punch to a gentle breeze!
(c) Ah, the "what if" questions! So, you're wondering what kind of charge-to-mass ratio a particle would need to have for the gravitational force to be equal to the electrical force, huh?
Let's use the equations we discussed earlier. Setting the gravitational force equal to the electrical force, we have:
G * (m1 * m2 / r^2) = k * (q1 * q2 / r^2)
Since the distances and positions of the particles are the same, we can cancel those out. Rearranging, we find:
(q1 * q2) / (m1 * m2) = (G * k)
Now, the charge-to-mass ratio (q/m) for a particle is just the charge divided by the mass. Let's represent the charge-to-mass ratio for one particle as q/m_1 and for the other particle as q/m_2. Substituting these values and the corresponding charges and masses into the equation, we get:
(q/m_1 * q/m_2) = (G * k)
Simplifying further, we find that the charge-to-mass ratio for the particles must be equal to the ratio of the gravitational and electrostatic constants:
(q/m) = sqrt((G * k))
Therefore, if the magnitude of the gravitational force between the particles is equal to the magnitude of the electrical force between them, the charge-to-mass ratio for the particles would be equal to the square root of the ratio of the gravitational constant to the electrostatic constant.
To solve this problem, we can use Coulomb's Law to find the electrical force exerted by one proton on the other.
(a) Coulomb's Law states that the electrical force between two charged particles is given by the equation:
F = k * (q1 * q2) / r^2
where F is the electrical force, k is the electrostatic constant, q1 and q2 are the charges of the two particles, and r is the distance between them.
The electrostatic constant, k, has a value of approximately 9.0 × 10^9 N·m^2/C^2.
In this case, both protons have the same magnitude of charge, which is equal to the elementary charge (e) and has a value of approximately 1.6 × 10^-19 C.
The distance between the protons is given as 3.80 × 10^-10 m.
Substituting these values into the equation, we can calculate the electrical force:
F = (9.0 × 10^9 N·m^2/C^2) * [(1.6 × 10^-19 C)^2 / (3.80 × 10^-10 m)^2]
Calculating this expression will give us the magnitude of the electrical force in newtons (N).
(b) To compare the magnitude of the electrical force with the gravitational force exerted by one proton on the other, we need to calculate the gravitational force.
The magnitude of the gravitational force between two objects is given by the equation:
F_gravity = G * (m1 * m2) / r^2
where F_gravity is the gravitational force, G is the gravitational constant, m1 and m2 are the masses of the two objects, and r is the distance between them.
The gravitational constant, G, has a value of approximately 6.674 × 10^-11 N·m^2/kg^2.
The mass of a proton is approximately 1.67 × 10^-27 kg.
Using the same distance between the protons as before (3.80 × 10^-10 m), we can calculate the gravitational force:
F_gravity = (6.674 × 10^-11 N·m^2/kg^2) * [(1.67 × 10^-27 kg)^2 / (3.80 × 10^-10 m)^2]
Calculating this expression will give us the magnitude of the gravitational force in newtons (N).
To find the ratio of the electrical force to the gravitational force, divide the magnitude of the electrical force by the magnitude of the gravitational force.
(c) To find the charge-to-mass ratio of a particle for the gravitational force to be equal to the electrical force, we set the two forces equal to each other:
F_gravity = F_electric
G * (m1 * m2) / r^2 = k * (q1 * q2) / r^2
Since the distances and constants are the same, we can cancel them out:
(m1 * m2) = (q1 * q2)
Rearranging the equation, we can solve for the charge-to-mass ratio, (q/m):
(q/m) = (m1 * m2) / (q1 * q2)
Substituting the values for mass and charge of a proton, we can calculate the charge-to-mass ratio in 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 objects is directly proportional to the product of their charges and inversely proportional to the square of the distance between them. The formula for Coulomb's Law is:
F = k * (q1 * q2) / r^2
Where:
F is the electrical force
k is the electrostatic constant (k = 8.99 x 10^9 N*m^2/C^2)
q1 and q2 are the charges of the two protons
r is the distance between the two protons
(a) In this case, the charges of the protons are the same (as they are both protons), so q1 = q2 = e, where e is the elementary charge (e = 1.60 x 10^-19 C). The distance between the protons is given as 3.80 x 10^-10 m. Substituting these values into the formula, we can calculate the electrical force:
F = (8.99 x 10^9 N*m^2/C^2) * ((1.60 x 10^-19 C)^2) / (3.80 x 10^-10 m)^2
Calculating this expression will give you the electrical force in Newtons (N).
(b) To compare the magnitude of the electrical force with the magnitude of the gravitational force between the protons, we can divide the electrical force by the gravitational force. The magnitude of the gravitational force between two objects can be calculated using Newton's Law of Universal Gravitation:
F_gravity = G * (m1 * m2) / r^2
Where:
F_gravity is the gravitational force
G is the gravitational constant (approximately 6.674 x 10^-11 N*m^2/kg^2)
m1 and m2 are the masses of the two protons (each proton has a mass of approximately 1.67 x 10^-27 kg)
r is the distance between the protons
The ratio of the electrical force to the gravitational force is:
(electrical force / gravitational force) = (F / F_gravity)
Calculating this expression will give you the desired ratio.
(c) If the magnitude of the gravitational force between two particles is equal to the magnitude of the electrical force between them, we can set the equations for the two forces equal to each other:
k * (q1 * q2) / r^2 = G * (m1 * m2) / r^2
We know that q1 = q2 = e and m1 = m2, so we can substitute these into the equation:
k * (e * e) / r^2 = G * (m1 * m2) / r^2
Simplifying the equation further, we can solve for the charge-to-mass ratio:
e^2 = G * (m1 * m2) / k
Charge-to-mass ratio = (e^2) / (G * m1)
By substituting the known values of e, G, and m1, you can find the charge-to-mass ratio in C/kg.