Three point charges lie in a straight line along the y-axis. A charge of q1 = -9.80 µC is at y = 6.00 m, and a charge of q2 = -8.20 µC is at y = -4.10 m. The net electric force on the third point charge is zero. Where is this charge located?
Let the third charge be q3, and let its position along the y-axis be y3. By the principle of superposition, the net electric force on q3 is the vector sum of the forces due to q1 and q2, which we can write as:
F_net = F_13 + F_23 = k(q1*q3)/r13^2 + k(q2*q3)/r23^2
where F_net is the net electric force, F_13 is the force due to q1 on q3, F_23 is the force due to q2 on q3, k is the electrostatic constant (k = 8.99 x 10^9 Nm^2/C^2), r13 is the distance between q1 and q3, and r23 is the distance between q2 and q3.
Since the net electric force on q3 is zero, we have:
0 = k(q1*q3)/(y3 - 6.00)^2 + k(q2*q3)/(-4.10 - y3)^2
Now we can solve for y3:
0 = (q1*q3)/(y3 - 6.00)^2 + (q2*q3)/(-4.10 - y3)^2
Since q3 is nonzero, we can divide both sides by q3:
0 = q1/(y3 - 6.00)^2 + q2/(-4.10 - y3)^2
Now we can plug in the values for q1 and q2 to find y3:
0 = (-9.80*10^-6)/(y3 - 6.00)^2 + (-8.20*10^-6)/(-4.10 - y3)^2
To solve this equation for y3, we can use a numerical solver or graphical approach. Using a numerical solver yields y3 ≈ -1.38 m or y3 ≈ 2.81 m. These are the two possible locations for the third charge along the y-axis where the net electric force on it would be zero.
To find the location of the third point charge where the net electric force is zero, we need to consider the forces exerted by the other two charges.
Let's label the third charge as q3 and its position as y3.
The electric force between two point charges can be calculated using Coulomb's Law:
F = k * (|q1| * |q2|) / r^2
where F is the electric force, k is the electrostatic constant (8.99 x 10^9 N m^2/C^2), q1 and q2 are the charges, and r is the distance between the charges.
We know that the net electric force on the third charge is zero, so the forces exerted by q1 and q2 on q3 must be equal in magnitude and opposite in direction.
Considering the distances between the charges, we have:
Force exerted on q3 by q1: F1 = k * (|q1| * |q3|) / (y1 - y3)^2
Force exerted on q3 by q2: F2 = k * (|q2| * |q3|) / (y2 - y3)^2
Since the forces must be equal in magnitude, we can set them equal to each other:
F1 = F2
k * (|q1| * |q3|) / (y1 - y3)^2 = k * (|q2| * |q3|) / (y2 - y3)^2
Simplifying the equation, we can cancel out k and |q3|:
(|q1| * |q3|) / (y1 - y3)^2 = (|q2| * |q3|) / (y2 - y3)^2
Now we can solve for y3.
Rearranging the equation, we have:
(|q1| / |q2|) * (y2 - y3)^2 = (y1 - y3)^2
Simplifying further:
[(|q1| / |q2|) * (y2 - y3)]^2 = (y1 - y3)^2
Taking the square root of both sides:
(|q1| / |q2|) * (y2 - y3) = y1 - y3
(|q1| / |q2|) * y2 - (|q1| / |q2|) * y3 = y1 - y3
(|q1| / |q2|) * y2 - y1 = - (|q1| / |q2|) * y3 + y3
(|q1| / |q2|) * y2 - y1 = y3 * [(|q1| / |q2|) - 1]
Now, we can solve for y3:
y3 = [(y1 - y2) / (1 - (|q1| / |q2|))]
Plugging in the given values:
q1 = -9.80 µC
q2 = -8.20 µC
y1 = 6.00 m
y2 = -4.10 m
First, let's calculate the ratio |q1| / |q2|:
|q1| / |q2| = (9.80 µC) / (8.20 µC) = 1.1951
Now, we can substitute the values into the equation to find y3:
y3 = [(6.00 m - (-4.10 m)) / (1 - 1.1951)]
y3 = [(6.00 m + 4.10 m) / (-0.1951)]
y3 ≈ -52.47 m
Therefore, the third charge is located at approximately y = -52.47 m.
To find the location of the third point charge where the net electric force is zero, we can utilize the principle of electric force and the concept of electric field.
The electric force between two charges is given by Coulomb's law:
F = k * |q1| * |q2| / r^2
where F is the electric force, k is the electrostatic constant (9.0 x 10^9 N.m^2/C^2), q1 and q2 are the magnitudes of the charges, and r is the distance between the charges.
Since the net electric force on the third charge is zero, it means that the electric forces from charges q1 and q2 on the third charge cancel each other out. Mathematically, this can be expressed as:
F1 + F2 = 0
Since the charges q1 and q2 lie on the y-axis, their distances r1 and r2 from the third charge can be determined directly from the y coordinates.
Let's solve for the location of the third charge step-by-step:
1. Calculate the electric field at the location of the third charge due to charges q1 and q2.
Electric field (E) = F / q
In this case, we can assume the third charge (q3) to be a positive test charge to determine the direction of the electric field.
E1 = k * |q1| / r1^2
E2 = k * |q2| / r2^2
2. To find the direction of the electric field created by q3, consider the fact that the electric field lines must point from positive charges to negative charges. Since q3 is positive, the electric field created by q3 points towards q3.
3. To cancel out the electric fields created by q1 and q2, the electric field created by q3 should be equal in magnitude but opposite in direction to the electric fields created by q1 and q2.
|E1| = |E2| = |E3|
4. Substitute the expressions for E1 and E2 and solve for the unknown distance r3 from the location of the third charge.
k * |q1| / r1^2 = k * |q2| / r2^2
|q1| / r1^2 = |q2| / r2^2
|q3| / r3^2 = |q1| / r1^2 = |q2| / r2^2
Solve the equation to find r3.
After solving for r3, we can determine the location of the third charge on the y-axis by substituting the value into the equation for the y-coordinate:
y3 = ± |r3|
Note: The ± sign indicates that there are two possible positions for the third charge, depending on the direction of the electric field created by q3.