trying to study but stuck with this one.
Two tiny spheres have the same mass and carry charges of the same magnitude. The mass of each sphere is 4.62 x 10-6 kg. The gravitational force that each sphere exerts on the other is balanced by the electric force. Determine the charge magnitude.
F(el)=k•q²/r²
k =9•10^9 N•m²/C²
F(gr) =G•m²/r²
G =6.67•10^-11 N•m²/kg²,
k•q²/r² = G•m²/r²,
q =sqrt(G•m²/k)
To solve this problem, we can set up an equation by equating the gravitational force and the electric force.
Step 1: Write down the given information:
- Mass of each sphere: 4.62 x 10^-6 kg
Step 2: Define the variables:
- Let q be the charge magnitude (in Coulombs) of each sphere.
Step 3: Write down the formulas for gravitational force and electric force:
- The gravitational force between two masses m1 and m2 is given by the formula:
F_grav = (G * m1 * m2) / r^2
Where G is the gravitational constant (6.67430 x 10^-11 N m^2/kg^2) and r is the distance between the masses.
- The electric force between two charges q1 and q2, separated by a distance r, is given by Coulomb's Law:
F_electric = (k * |q1 * q2|) / r^2
Where k is the electrostatic constant (8.988 x 10^9 N m^2/C^2).
Step 4: Set up the equation:
Since the masses of the spheres are equal, the gravitational force between them will be the same for each sphere. Therefore, to balance the forces, the magnitudes of the electric charges must be the same. Thus, we can equate the two forces:
(G * m^2) / r^2 = (k * q^2) / r^2
Step 5: Cancel out the common terms and rearrange the equation:
G * m^2 = k * q^2
Step 6: Solve for the charge magnitude:
q^2 = (G * m^2) / k
q = sqrt((G * m^2) / k)
Step 7: Substitute the given values and calculate:
q = sqrt((6.67430 x 10^-11 N m^2/kg^2 * (4.62 x 10^-6 kg)^2) / (8.988 x 10^9 N m^2/C^2))
q ≈ sqrt(1.96 x 10^-24 / 8.0958 x 10^-23)
q ≈ sqrt(0.024158)
q ≈ 0.155 C
Therefore, the magnitude of each sphere's charge is approximately 0.155 C.
To determine the magnitude of the charge on each sphere, we need to set up equations to equate the gravitational and electric forces.
Let's start with the gravitational force. The gravitational force between two objects (Fg) is given by the equation:
Fg = G * ((mass1 * mass2) / distance^2),
where G is the universal gravitational constant (6.67430 x 10^-11 Nm^2/kg^2), and distance is the separation between the centers of the two spheres.
Now, let's move on to the electric force. The electric force between two charged objects (Fe) is given by the equation:
Fe = k * ((charge1 * charge2) / distance^2),
where k is the Coulomb's constant (8.988 × 10^9 Nm^2/C^2).
In this case, since the charges on the two spheres have the same magnitude, we can set the charges as q and -q, respectively.
Now, we can set up the equations to equate the gravitational and electric forces:
G * ((mass1 * mass2) / distance^2) = k * ((charge1 * charge2) / distance^2).
Since the masses of the spheres are the same (4.62 x 10^-6 kg), we have:
G * ((mass1 * mass2) / distance^2) = k * ((charge * -charge) / distance^2).
We can simplify this equation to:
G * (mass^2) = k * (charge^2).
Next, substitute the values for G, mass, and k:
(6.67430 x 10^-11 Nm^2/kg^2) * ((4.62 x 10^-6 kg)^2) = (8.988 × 10^9 Nm^2/C^2) * (charge^2).
Now, solve for the charge:
charge^2 = ((6.67430 x 10^-11 Nm^2/kg^2) * ((4.62 x 10^-6 kg)^2)) / (8.988 × 10^9 Nm^2/C^2).
By calculating this expression, you will obtain the value of charge^2. To find the magnitude of the charge, take the square root of the obtained value.
Remember to keep track of the units while performing the calculations.