Two tiny spheres have the same mass and carry charges of the same magnitude. The mass of each sphere is 3.30 10-6 kg. The gravitational force that each sphere exerts on the other is balanced by the electric force.
(a) What algebraic signs can the charges have?
1the same algebraic sign, both positive or both negative
opposite algebraic sign, one positive and one negative
the same algebraic sign, both negative
the same algebraic sign, both positive
(b) Determine the charge magnitude.
(a) The two forces are balanced, so the algebraic signs of the charges must be opposite. Therefore, one sphere must have a positive charge and the other must have a negative charge.
(b) To determine the charge magnitude, we can equate the gravitational force and the electric force:
Gravitational force (Fg) = Electric force (Fe)
Using Newton's law of universal gravitation, the gravitational force between the two spheres can be expressed as:
Fg = G * (m1 * m2) / r^2
Where G is the gravitational constant, m1 and m2 are the masses of the spheres, and r is the distance between their centers.
The electric force between the two spheres can be expressed as:
Fe = k * (q1 * q2) / r^2
Where k is the Coulomb constant, q1 and q2 are the charges on the spheres, and r is the distance between their centers.
Since the masses and distances are the same for both spheres, we can set Fg = Fe and solve for the charge magnitude:
G * (m1 * m2) / r^2 = k * (q1 * q2) / r^2
Simplifying, we get:
G * m1 * m2 = k * q1 * q2
Plugging in the given values:
(6.67 * 10^-11 N*m^2/kg^2) * (3.30 * 10^-6 kg) * (3.30 * 10^-6 kg) = (9 * 10^9 N*m^2/C^2) * (q1 * q2)
Solving for q1 * q2, we find:
q1 * q2 = [(6.67 * 10^-11 N*m^2/kg^2) * (3.30 * 10^-6 kg) * (3.30 * 10^-6 kg)] / (9 * 10^9 N*m^2/C^2)
Calculating this expression gives us the magnitude of the charges.
In order to answer both (a) and (b), we need to consider the forces involved and set them equal to each other.
(a) The gravitational force between the two spheres can be calculated using Newton's law of universal gravitation:
F_grav = G * (m1 * m2) / r^2
Here, G represents the gravitational constant, m1 and m2 are the masses of the spheres, and r is the distance between them. Since the masses are the same, this equation simplifies to:
F_grav = G * (m^2) / r^2
On the other hand, the electric force between the spheres can be calculated using Coulomb's law:
F_electric = k * (|q1 * q2|) / r^2
Here, k is the electrostatic constant, q1 and q2 are the charges on the spheres, and r is the distance between them.
Since the gravitational force is balanced by the electric force, we set them equal to each other:
F_grav = F_electric
G * (m^2) / r^2 = k * (|q1 * q2|) / r^2
Simplifying further:
G * (m^2) = k * (|q1 * q2|)
Now, let's consider the possible signs for the charges:
Case 1: q1 and q2 both positive
In this case, the absolute value of their product |q1 * q2| will be positive.
Case 2: q1 and q2 both negative
In this case, the absolute value of their product |q1 * q2| will also be positive.
Case 3: q1 positive and q2 negative, or vice versa
In this case, the absolute value of their product |q1 * q2| will be negative.
Based on the equation G * (m^2) = k * (|q1 * q2|), we see that the right side must also be positive to maintain equality. Therefore, the correct answer for (a) is: the same algebraic sign, both positive.
(b) To find the charge magnitude, let's substitute the known values into the equation G * (m^2) = k * (|q1 * q2|):
(6.67430 * 10^-11 N(m/kg)^2) * (3.30 * 10^-6 kg)^2 = (8.988 * 10^9 N(m/C)^2) * (|q1 * q2|)
Simplifying further:
1.456 * 10^-16 N(m/kg)^2 = (|q1 * q2|)
Since the masses and constants are known, we can solve for |q1 * q2|:
|q1 * q2| = 1.456 * 10^-16 N(m/kg)^2
To find the charge magnitude, we take the square root of |q1 * q2|:
|q1| = sqrt(1.456 * 10^-16 N(m/kg)^2)
Calculating this value gives us the charge magnitude.