We can use Coulomb's Law to find the position on the x-axis where a third charge should be positioned to experience no force.
Coulomb's Law states that the force between two charges is given by:
F = k * (q1 * q2) / r^2
Where:
- F is the force between the charges,
- k is the Coulomb's constant (9.0 x 10^9 N*m^2/C^2),
- q1 and q2 are the magnitudes of the charges,
- r is the distance between the charges.
To find the position on the x-axis where the third charge should be placed, we need to consider the forces acting on it. Let's assume the third charge is q3 and its position is x3. If q3 experiences no force, the forces on it from q1 and q2 should cancel out.
The force on q3 due to q1 is:
F1 = k * (q1 * q3) / (x3)^2
The force on q3 due to q2 is:
F2 = k * (q2 * q3) / (10.0 cm - x3)^2
For the net force on q3 to be zero, we must have F1 = -F2. Since the charges are of the same magnitude (1.00 μC and 2.00 µC), we can set up the equation as:
k * (q1 * q3) / (x3)^2 = -k * (q2 * q3) / (10.0 cm - x3)^2
Cancelling the constants and converting the distances to meters:
(1.00 * 10^-6 C * q3) / (x3)^2 = -(2.00 * 10^-6 C * q3) / (0.10 m - x3)^2
Now we can solve this equation to find the value of x3. Let's multiply both sides by (x3)^2 and (0.10 m - x3)^2 to eliminate the denominators:
(1.00 * 10^-6 C * q3) * (0.10 m - x3)^2 = -(2.00 * 10^-6 C * q3) * (x3)^2
Expanding and rearranging:
(1.00 * 10^-6 C * q3) * (0.01 m^2 - 0.20 m * x3 + (x3)^2) = -(2.00 * 10^-6 C * q3) * (x3)^2
Multiplying out both sides:
0.01 m^2 * (1.00 * 10^-6 C * q3) - 0.20 m * (1.00 * 10^-6 C * q3) * x3 + (x3)^2 * (1.00 * 10^-6 C * q3) = -(2.00 * 10^-6 C * q3) * (x3)^2
Simplifying:
0.01 m^2 * (1.00 * 10^-6 C * q3) - 0.20 m * (1.00 * 10^-6 C * q3) * x3 + (x3)^2 * (1.00 * 10^-6 C * q3) + (2.00 * 10^-6 C * q3) * (x3)^2 = 0
Combining like terms:
(3.00 * 10^-6 C * q3) * (x3)^2 - 0.20 m * (1.00 * 10^-6 C * q3) * x3 + 0.01 m^2 * (1.00 * 10^-6 C * q3) = 0
This is a quadratic equation in terms of x3, which we can solve to find the values of x3 where the net force on q3 is zero.
Now, let's substitute the values of the charges:
q1 = 1.00 μC = 1.00 * 10^-6 C
q2 = -2.00 μC = -2.00 * 10^-6 C
And the values of the lengths:
x1 = 0 (as it is fixed at the origin)
x2 = 10.0 cm = 10.0 * 10^-2 m
Substituting the values into the equation and solving, we get the values of x3 where q3 experiences no force.
A positive charge q = 1.00 μC is fixed at the origin, and a second charge q2=-2.00 µC is fixed at x = 10.0 cm. Where along the x-axis should a third charge be positioned so that it experiences no force?
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