Two charges, -17 and +3.6 µC, are fixed in place and separated by 3.2 m.
(a) At what spot along a line through the charges is the net electric field zero? Locate this spot relative to the positive charge. (Hint: The spot does not necessarily lie between the two charges.)
(b) What would be the force on a charge of +1 µC placed at this spot?
To find the spot along the line between the two charges where the net electric field is zero, we can make use of the fact that the electric field due to a point charge is given by Coulomb's law.
(a) The net electric field at a point is the vector sum of the electric fields due to each charge individually. At the spot where the net electric field is zero, the magnitudes of the electric fields due to each charge must be equal.
Let's assume that the positive charge (+3.6 µC) is located at the origin (0,0) on the coordinate system, while the negative charge (-17 µC) is located at position (3.2 m, 0).
We need to find the coordinates (x,0) of the spot along the line where the net electric field is zero.
The electric field due to the negative charge at a point (x,0) is given by:
E_negative = k * |q_negative| / r_negative²
where k is the Coulomb's constant (8.99 x 10^9 N·m²/C²), |q_negative| is the magnitude of the negative charge (-17 µC), and r_negative is the distance between the negative charge and the spot (x,0).
Similarly, the electric field due to the positive charge at the same point is given by:
E_positive = k * |q_positive| / r_positive²
where |q_positive| is the magnitude of the positive charge (+3.6 µC), and r_positive is the distance between the positive charge and the spot (x,0).
To find the spot where the net electric field is zero, we need to set these two electric fields equal to each other:
E_positive = E_negative
k * |q_positive| / r_positive² = k * |q_negative| / r_negative²
Simplifying the equation:
|q_positive| / r_positive² = |q_negative| / r_negative²
Substituting the known values:
(3.6 x 10^-6 C) / r_positive² = (-17 x 10^-6 C) / r_negative²
Solving for r_positive²:
r_positive² = r_negative² * (|q_positive| / |q_negative|)
Substituting the known values:
r_positive² = (3.2 m)² * (3.6 x 10^-6 C) / (17 x 10^-6 C)
Simplifying:
r_positive² = (3.2 m)² * (0.212)
r_positive² = 2.1536
Taking the square root of both sides:
r_positive ≈ 1.468 m
Therefore, the distance from the positive charge (+3.6 µC) to the spot where the net electric field is zero is approximately 1.468 m.
To find the spot relative to the positive charge, we take the negative of the value:
Spot relative to positive charge = -1.468 m
(b) The force on a charge of +1 µC placed at the spot can be calculated using Coulomb's law:
F = k * |q1| * |q2| / r²
where F is the force, k is the Coulomb's constant, |q1| and |q2| are the magnitudes of the charges, and r is the distance between the charges.
In this case, the force on the +1 µC charge can be calculated using the equation:
F = k * |q1| * |q2| / r²
F = (8.99 x 10^9 N·m²/C²) * (1 x 10^-6 C) * (3.6 x 10^-6 C) / (1.468 m)²
Simplifying the calculation, we have:
F ≈ 2.481 x 10^-3 N
Therefore, the force on a charge of +1 µC placed at the spot where the net electric field is zero is approximately 2.481 x 10^-3 N.