Two point charges lie along the y axis. A charge of q1 = -12.0 µC is at y = 6.0 m, and a charge of q2 = -5.0 µC is at y = -3.0 m. Locate the point (other than infinity) at which the total electric field is zero.
The are 9 meters apart. So to make it easier, we shift y axis up by adding 3.0, we will subtract it later.
k(-5)/x^2=k(-12/(9-x)^2
x is the distance from the lower charge.
12x^2=5(81+x^2-18x)
solve for x. Then add -3.
To find the point where the total electric field is zero, we need to calculate the electric field contributions from each charge and find the point where the sum of these contributions is zero.
Let's assume the point at which the total electric field is zero is located at y = d.
The electric field contribution from q1 can be calculated using the formula:
E1 = (k * q1) / r1^2
where k is the electrostatic constant (k = 8.99 x 10^9 Nm^2/C^2), q1 is the charge of q1 (-12.0 µC), and r1 is the distance between q1 and the point (y = d).
The electric field contribution from q2 can be calculated using the same formula:
E2 = (k * q2) / r2^2
where q2 is the charge of q2 (-5.0 µC), and r2 is the distance between q2 and the point (y = d).
Since the point at which the total electric field is zero, we can equate the electric field contributions:
E1 + E2 = 0
Substituting the formulas for E1 and E2:
(k * q1) / r1^2 + (k * q2) / r2^2 = 0
Simplifying the equation:
(k * q1) / r1^2 = -(k * q2) / r2^2
Cross multiplying:
r2^2 * q1 + r1^2 * q2 = 0
Since we are solving for d, let's express r1 and r2 in terms of d:
r1 = d - 6.0 m
r2 = d + 3.0 m
Substituting these values into the equation:
(d - 6.0 m)^2 * q1 + (d + 3.0 m)^2 * q2 = 0
Now we can solve this equation for d to find the point at which the total electric field is zero.
To locate the point where the total electric field is zero, we need to find a point along the y-axis where the electric field due to q1 is equal in magnitude but opposite in direction to the electric field due to q2.
The electric field created by a point charge can be calculated using the formula:
E = k * |q| / r^2
where E is the electric field, k is the Coulomb's constant (9.0 x 10^9 N m^2/C^2), |q| is the magnitude of the charge, and r is the distance between the charge and the point where the electric field is being calculated.
Let's call the point along the y-axis where the electric field is zero as y0. At this point, the electric field created by q1 and q2 will be:
E1 = k * |q1| / (y0 - 6)^2
E2 = k * |q2| / (y0 + 3)^2
Since we want the total electric field to be zero, we can set E1 equal to the negative of E2:
E1 = -E2
k * |q1| / (y0 - 6)^2 = -k * |q2| / (y0 + 3)^2
Now, let's plug in the given values:
q1 = -12.0 µC = -12.0 x 10^-6 C
q2 = -5.0 µC = -5.0 x 10^-6 C
We can simplify the equation by canceling out k and rearranging the terms:
|q1| / (y0 - 6)^2 = |q2| / (y0 + 3)^2
Squaring both sides:
(q1)^2 / (y0 - 6)^4 = (q2)^2 / (y0 + 3)^4
Plugging in the values:
(12.0 x 10^-6)^2 / (y0 - 6)^4 = (5.0 x 10^-6)^2 / (y0 + 3)^4
Now, we can cross-multiply and solve for y0 to find the point where the total electric field is zero.