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?
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.
To find the position along the x-axis where a third charge should be placed in order to experience no force, we can use the principle of electrostatic equilibrium. In this case, the net force on the third charge should be zero.
Given:
Charge at the origin (q1) = 1.00 μC
Charge at x = 10.0 cm (q2) = -2.00 µC
Let's assume the charge at the third position (q3) is placed at x = d.
According to Coulomb's Law, the force between two charges is given by:
F = (k * |q1 * q2|) / r^2
Where:
k = Coulomb's constant = 9.0 x 10^9 Nm^2/C^2
|q1 * q2| = magnitude of the product of the charges
r = distance between the charges
Since we want the net force on the third charge to be zero, the net force equation can be written as:
F(net) = F1 + F2
The force due to q1 on q3 (F1) is given by:
F1 = (k * |q1 * q3|) / r1^2
The force due to q2 on q3 (F2) is given by:
F2 = (k * |q2 * q3|) / r2^2
Since F(net) = 0, we can write the equation as:
F1 + F2 = 0
Substituting the expressions for F1 and F2, we get:
(k * |q1 * q3|) / r1^2 + (k * |q2 * q3|) / r2^2 = 0
Plugging in the given values:
k = 9.0 x 10^9 Nm^2/C^2
q1 = 1.00 μC = 1.00 x 10^-6 C
q2 = -2.00 µC = -2.00 x 10^-6 C
r1 = d
r2 = 10.0 cm = 10^-1 m
The equation becomes:
(9.0 x 10^9 Nm^2/C^2 * |(1.00 x 10^-6 C) * q3|) / d^2 + (9.0 x 10^9 Nm^2/C^2 * |(-2.00 x 10^-6 C) * q3|) / (10^-1 m)^2 = 0
Simplifying further:
(9.0 x 10^9 Nm^2/C^2 * q3) / d^2 - (18.0 x 10^9 Nm^2/C^2 * q3) / (10^-2 m)^2 = 0
Combining the terms:
(9.0 x 10^9 Nm^2/C^2 * q3 / d^2) - (18.0 x 10^9 Nm^2/C^2 * q3 / (10^-2 m)^2) = 0
Multiplying through by (10^-2 m)^2:
(9.0 x 10^9 Nm^2/C^2 * q3 * (10^-2 m)^2) / d^2 - 18.0 = 0
Simplifying:
(9.0 x 10^7 Nm^2/C^2 * q3) / d^2 - 18.0 = 0
Rearranging the equation, we get:
(9.0 x 10^7 Nm^2/C^2 * q3) = 18.0 * d^2
Solving for d^2:
d^2 = (9.0 x 10^7 Nm^2/C^2 * q3) / 18.0
Now, substituting the charge value for q3:
d^2 = (9.0 x 10^7 Nm^2/C^2 * q3) / 18.0
d^2 = (9.0 x 10^7 Nm^2/C^2 * 1.00 x 10^-6 C) / 18.0
Calculating the value:
d^2 = 0.05 m^2
Taking the square root of both sides:
d = ±0.2236 m
Since we are looking for a positive distance, the third charge should be positioned at x = 0.2236 meters along the positive x-axis in order to experience no force.
To find the position along the x-axis where the third charge experiences no force, we can use the principle of electrostatic equilibrium. In this case, the net force acting on the third charge must be zero.
The force between two charges is given by Coulomb's law:
F = k * q1 * q2 / r^2
Where:
F is the force between the charges,
k is the Coulomb's constant (k = 9.0 x 10^9 N·m²/C²),
q1 and q2 are the magnitudes of the charges, and
r is the distance between the charges.
In this problem, the force between the first and third charges is repulsive (since they have the same sign), and the force between the second and third charges is attractive (since they have opposite signs). We want the net force on the third charge to be zero, so we need to set up the condition:
F1 + F2 = 0
Let's break down the problem:
1. Calculate the force exerted by the positive charge at the origin (F1):
- q1 = 1.00 μC = 1.00 x 10^-6 C
- q3 = unknown (let's call it x)
- r1 = distance between the charges = x (since the third charge will be positioned at x)
- Plugging these values into Coulomb's law: F1 = k * q1 * q3 / r1^2
2. Calculate the force exerted by the negative charge at x = 10.0 cm (F2):
- q2 = -2.00 μC = -2.00 x 10^-6 C
- r2 = distance between the charges = 0.10 m (since x = 10.0 cm)
- Plugging these values into Coulomb's law: F2 = k * q2 * q3 / r2^2
3. Set up the equation F1 + F2 = 0 and solve for x:
- Substitute the values of F1 and F2 into the equation: k * q1 * q3 / r1^2 + k * q2 * q3 / r2^2 = 0
- Rearrange the equation and solve for x.
By following these steps, you can calculate the position along the x-axis where a third charge should be positioned so that it experiences no force.