1.) A small sphere of charge 2.4 μC experience a force of 0.36 N when a second sphere of unknown charge is placed 5.5 cm from it. What is the charge of the second sphere?
2.) Two identically charged sphere placed 12 cm apart have an electric force of0.28 N between them. What is the charge of each sphere?
3.) In an experience using Coulomb's apparatus, a sphere with a charge of 3.6x10^-8 C is 1.4 cm from a second sphere of unknown charge. The force between the spheres is 2.7x10^-2 N. What is the charge of the second sphere?
4.) The force between a proton and an electron is 3.5x10^-10 N. What is the distance between these two particles
1.) To find the charge of the second sphere, we can use Coulomb's law, which states that the force between two charges is proportional to the product of the charges and inversely proportional to the square of the distance between them.
Coulomb's law can be written as:
F = k * (q1 * q2) / r^2
Where:
F is the force between the charges,
k is the electrostatic constant (k ≈ 9 × 10^9 N m^2/C^2),
q1 and q2 are the charges of the two spheres, and
r is the distance between them.
Given:
Charge of the first sphere (q1) = 2.4 μC = 2.4 × 10^-6 C
Force (F) = 0.36 N
Distance (r) = 5.5 cm = 0.055 m
Plugging these values into Coulomb's law, we can solve for the charge (q2) of the second sphere:
0.36 N = (9 × 10^9 N m^2/C^2) * ((2.4 × 10^-6 C) * q2) / (0.055 m)^2
To find q2, we can rearrange the equation:
q2 = (0.36 N * (0.055 m)^2) / ((9 × 10^9 N m^2/C^2) * (2.4 × 10^-6 C))
= 3.1356 × 10^-11 C
Therefore, the charge of the second sphere is approximately 3.1356 × 10^-11 C.
2.) To find the charge of each sphere, we can use Coulomb's law, similar to the previous question.
Given:
Force (F) = 0.28 N
Distance (r) = 12 cm = 0.12 m
Using the same equation as before:
F = k * (q1 * q2) / r^2
We know that both spheres have the same charge, so q1 = q2 = q (let's call it q).
0.28 N = (9 × 10^9 N m^2/C^2) * (q * q) / (0.12 m)^2
To find q, we can rearrange the equation:
q^2 = (0.28 N * (0.12 m)^2) / ((9 × 10^9 N m^2/C^2))
= 3.2 × 10^-11 C
Taking the square root of both sides:
q = √(3.2 × 10^-11 C)
= 5.65 × 10^-6 C
Therefore, the charge of each sphere is approximately 5.65 × 10^-6 C.
3.) Again, we can use Coulomb's law to find the charge of the second sphere.
Given:
Charge of the first sphere (q1) = 3.6 × 10^-8 C
Force (F) = 2.7 × 10^-2 N
Distance (r) = 1.4 cm = 0.014 m
Using the same equation as before:
F = k * (q1 * q2) / r^2
Let's solve for q2:
2.7 × 10^-2 N = (9 × 10^9 N m^2/C^2) * ((3.6 × 10^-8 C) * q2) / (0.014 m)^2
To find q2, rearrange the equation:
q2 = (2.7 × 10^-2 N * (0.014 m)^2) / ((9 × 10^9 N m^2/C^2) * (3.6 × 10^-8 C))
= 1 × 10^-10 C
Therefore, the charge of the second sphere is 1 × 10^-10 C.
4.) To find the distance between the proton and electron, we can use Coulomb's law once again.
Given:
Force (F) = 3.5 × 10^-10 N
Using the same equation as before:
F = k * (q1 * q2) / r^2
We know that the proton and electron have charges of opposite signs, so q1 = e and q2 = -e, where e is the elementary charge (e ≈ 1.6 × 10^-19 C).
Plugging in the values:
3.5 × 10^-10 N = (9 × 10^9 N m^2/C^2) * ((e * -e) / r^2)
Simplifying the equation:
3.5 × 10^-10 N = - (9 × 10^9 N m^2/C^2) * (e^2 / r^2)
Since e^2 is a constant and k is a constant, we can write:
3.5 × 10^-10 N = C / r^2
Where C is a constant.
To find r, rearrange the equation:
r^2 = C / (3.5 × 10^-10 N)
r = √(C / (3.5 × 10^-10 N))
The constant C will depend on the charges and the value of k, but we can't determine the exact value of r with the given information.
To solve these problems, we can use Coulomb's Law, which states that the electric 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 electric force between the charges, k is the electrostatic constant (k = 9.0 x 10^9 N m^2/C^2), q1 and q2 are the charges of the two objects, and r is the distance between them.
Now let's solve the problems step by step:
1.) To find the charge of the second sphere, we know the force (F = 0.36 N), the charge of the first sphere (q1 = 2.4 μC), and the distance between them (r = 5.5 cm = 0.055 m). We can rearrange Coulomb's Law to solve for the charge of the second sphere (q2):
F = k * (q1 * q2) / r^2
Rearrange:
q2 = (F * r^2) / (k * q1)
Substitute the given values:
q2 = (0.36 N * (0.055 m)^2) / (9.0 x 10^9 N m^2/C^2 * 2.4 x 10^-6 C)
Calculate:
q2 = 0.0444 C or 44.4 μC
Therefore, the charge of the second sphere is 44.4 μC.
2.) To find the charge of each sphere, we know the force (F = 0.28 N), the distance between them (r = 12 cm = 0.12 m), and that they are identical charges, which means they have the same charge (q1 = q2 = q).
Using Coulomb's Law:
F = k * (q1 * q2) / r^2
Since q1 = q2 = q, we can rewrite the equation as:
F = k * (q * q) / r^2
Solve for q:
q^2 = (F * r^2) / k
Take the square root:
q = sqrt((F * r^2) / k)
Substitute the given values:
q = sqrt((0.28 N * (0.12 m)^2) / (9.0 x 10^9 N m^2/C^2))
Calculate:
q = 4.13 x 10^-7 C or 413 nC
Therefore, the charge of each sphere is 413 nC.
3.) To find the charge of the second sphere, we know the force (F = 2.7 x 10^-2 N), the charge of the first sphere (q1 = 3.6 x 10^-8 C), and the distance between them (r = 1.4 cm = 0.014 m).
Using Coulomb's Law (similar to Problem 1):
F = k * (q1 * q2) / r^2
Rearrange:
q2 = (F * r^2) / (k * q1)
Substitute the given values:
q2 = (2.7 x 10^-2 N * (0.014 m)^2) / (9.0 x 10^9 N m^2/C^2 * 3.6 x 10^-8 C)
Calculate:
q2 = 1.09 x 10^-7 C or 109 nC
Therefore, the charge of the second sphere is 109 nC.
4.) To find the distance between a proton and an electron when the force between them is given, we know the force (F = 3.5 x 10^-10 N). Since the question doesn't mention the charges of the particles, we can assume that it's a proton (q1 = 1.6 x 10^-19 C) and an electron (q2 = -1.6 x 10^-19 C) because they are commonly paired in this context.
Using Coulomb's Law (similar to Problems 1 and 3):
F = k * (q1 * q2) / r^2
Rearrange:
r^2 = (k * (q1 * q2)) / F
Take the square root:
r = sqrt((k * (q1 * q2)) / F)
Substitute the given values:
r = sqrt((9.0 x 10^9 N m^2/C^2 * (1.6 x 10^-19 C * -1.6 x 10^-19 C)) / (3.5 x 10^-10 N))
Calculate:
r = 4.88 x 10^-11 m or 48.8 pm
Therefore, the distance between the proton and the electron is 48.8 picometers.
These are all Coulomb's law
F = k Q1 Q2/r^2
Just plug in but watch units
2.4*10^-6 Coulombs at .055 meters in prob 1 for example.
k = 9*10^9
e = 1.6*10^-19 Coulombs, negative for electron, positive for proton