Two small nonconducting spheres have a total charge of 91.7 µC.
(a) When placed 1.10 m apart, the force each exerts on the other is 12.0 N and is repulsive. What is the charge on each? (Round your answers to the nearest 0.1 µC.)
(b) What if the force were attractive?
k =9•10^9 N•m²/C²
F=k•q1•q2/r² = k•q ²/r²,
q =sqrt(F• r²/k) = ...
100 % incorrect!
To find the charge on each small nonconducting sphere, we can use Coulomb's law, which states that the force between two charged objects is proportional to the product of their charges and inversely proportional to the square of the distance between them.
(a) Given:
Total charge on both spheres = 91.7 µC
Distance between the spheres = 1.10 m
Force between the spheres = 12.0 N (repulsive)
Let's assume the charges on the spheres are q1 and q2, with q1 being the charge on the first sphere and q2 being the charge on the second sphere.
Using Coulomb's law, we can write the equation for the force between the spheres as:
F = k * (q1 * q2) / r^2
Where:
F = force between the spheres (12.0 N)
k = Coulomb's constant (9 x 10^9 N m^2/C^2)
q1 = charge on the first sphere
q2 = charge on the second sphere
r = distance between the spheres (1.10 m)
Substituting the given values into the equation, we get:
12.0 N = (9 x 10^9 N m^2/C^2) * (q1 * q2) / (1.10 m)^2
Now, we know that the total charge on both spheres is 91.7 µC. So, q1 + q2 = 91.7 µC.
We can solve these two equations to find the values of q1 and q2.
Using a numerical solver or manually rearranging the equations and solving, we find that q1 ≈ 31.9 µC and q2 ≈ 59.8 µC.
Therefore, the charge on the first sphere (q1) is approximately 31.9 µC, and the charge on the second sphere (q2) is approximately 59.8 µC.
(b) If the force were attractive, the charges on the spheres would have opposite signs. In this case, the charges would be -31.9 µC and -59.8 µC, respectively.
To solve this problem, we can use Coulomb's law, which states that the force between two charges is proportional to the product of their magnitudes and inversely proportional to the square of the distance between them. Coulomb's law formula is given by:
F = k * (q₁ * q₂) / r²
Where:
F is the force between the charges,
k is the electrostatic constant (k = 9 × 10^9 N m²/C²),
q₁ and q₂ are the charges on the spheres, and
r is the distance between the spheres.
Let's now solve part (a) of the question:
(a) The force between the two spheres is given as 12.0 N, and the distance between them is 1.10 m. We have to find the charge on each sphere.
Step 1: Plug the given values into the Coulomb's law formula:
12.0 N = (9 × 10^9 N m²/C²) * ((q₁ * q₂) / (1.10 m)²
Step 2: Simplify the equation:
12.0 N = (9 × 10^9 N m²/C²) * (q₁ * q₂) / (1.21 m²)
Step 3: Rearrange the equation to find the product of the charges:
(q₁ * q₂) = (12.0 N) * (1.21 m²) / (9 × 10^9 N m²/C²)
Step 4: Calculate the product of the charges:
(q₁ * q₂) = 1.628 N m²/C²
Since the two spheres have the same charge, we can let q₁ = q₂ = q. Therefore:
(q * q) = 1.628 N m²/C²
q² = 1.628 N m²/C²
q = √(1.628 N m²/C²)
q ≈ 1.28 µC
So, the charge on each sphere is approximately 1.3 µC.
Now let's solve part (b) of the question:
(b) If the force were attractive, the magnitude of the force would remain the same. Therefore, the charge on each sphere would still be approximately 1.3 µC. The signs of the charges, however, would be opposite, one positive and the other negative, since opposite charges attract each other.
Therefore, if the force were attractive, the charge on one sphere would be approximately +1.3 µC and the charge on the other sphere would be approximately -1.3 µC.