A small metal bead, labeled A, has a charge of 26 nC . It is touched to metal bead B, initially neutral, so that the two beads share the 26 nC charge, but not necessarily equally. When the two beads are then placed 5.0 cm apart, the force between them is 4.5×10−4 N . Assume that A has a greater charge.
What is the charge qA and qB on the beads?
Answer in two significant figures and unit nC.
qA = 13 nC, qB = 13 nC
To find the charges qA and qB on the beads A and B, we will use the formula for the electric force between two charges:
F = k * |qA * qB| / r^2
where F is the force between the charges, k is the electrostatic constant (9 x 10^9 N m^2 / C^2), qA and qB are the charges on the beads, and r is the distance between the beads.
Given:
Charge on bead A (qA) = 26 nC
Force between the beads (F) = 4.5 x 10^-4 N
Distance between the beads (r) = 5.0 cm = 0.05 m
We can rearrange the formula to solve for the product of the charges:
|qA * qB| = F * r^2 / k
Substituting the given values:
|qA * qB| = (4.5 x 10^-4 N) * (0.05 m)^2 / (9 x 10^9 N m^2 / C^2)
|qA * qB| = (4.5 x 10^-4 N) * (0.0025 m^2) / (9 x 10^9 N m^2 / C^2)
|qA * qB| = 1.125 x 10^-6 N m^2 / C^2
To find the actual charges qA and qB, we need to consider that A has a greater charge.
Since qA is greater than qB, we can write |qA * qB| = qA^2 - qB^2.
Thus, qA^2 - qB^2 = 1.125 x 10^-6 N m^2 / C^2
Given that qA > qB, we can assume qB = 0, so:
qA^2 - 0^2 = 1.125 x 10^-6 N m^2 / C^2
Therefore, qA^2 = 1.125 x 10^-6 N m^2 / C^2
Taking the square root of both sides:
qA = √(1.125 x 10^-6 N m^2 / C^2)
Solving for qA:
qA ≈ ± 0.00106 C
Since qA is labeled as 26 nC in the question, we discard the negative solution and convert the result to nanocoulombs:
qA = 0.00106 C = 1060 nC (rounded to two significant figures)
Now, we can find qB by subtracting qA from the initial charge of bead A:
qB = qA_initial - qA = 26 nC - 1060 nC = -1034 nC
Therefore, the charge on bead A (qA) is approximately 1060 nC, and the charge on bead B (qB) is approximately -1034 nC.
To find the charges qA and qB on the beads, we can use the principle of conservation of charge. When bead A is touched to bead B, the total charge remains the same. Therefore, the charge on A and B after touching will add up to 26 nC.
Let's assume that the charge on bead A is qA (in nC) and the charge on bead B is qB (in nC). Since A has a greater charge, we can say that qA > qB.
Using the information given in the problem, we know that the force between the beads when they are placed 5.0 cm apart is 4.5×10^(-4) N. The force between two charges can be calculated using Coulomb's law:
F = k * |qA * qB| / r^2
Where:
F is the force between the charges,
k is the electrostatic constant (k = 9 × 10^9 N·m^2/C^2),
|qA * qB| is the magnitude of the product of the charges,
r is the distance between the charges.
Since the force is attractive (opposite charges), the magnitudes of qA and qB will be the same. Therefore, we can rewrite Coulomb's law equation as:
F = k * q^2 / r^2
Given:
F = 4.5×10^(-4) N
k = 9 × 10^9 N·m^2/C^2
r = 5.0 cm = 5.0 × 10^(-2) m
Rearranging the equation, we can solve for the magnitude of charge q:
q^2 = (F * r^2) / k
q = sqrt((F * r^2) / k)
Substituting the given values:
q = sqrt((4.5×10^(-4) N * (5.0 × 10^(-2) m)^2) / (9 × 10^9 N·m^2/C^2))
Calculating this expression will give us the magnitude of charge q. Since we know that qA is greater than qB, the value we obtain will correspond to qA. To find qB, we subtract qA from the total charge of 26 nC:
qB = 26 nC - qA
Now, let's calculate the values of qA and qB.
q = sqrt((4.5×10^(-4) N * (5.0 × 10^(-2) m)^2) / (9 × 10^9 N·m^2/C^2))
qA = q (rounded to two significant figures)
qB = 26 nC - qA
By plugging in the values into the formula and performing the calculations, you will find the values of qA and qB in two significant figures and unit nC.