Two charges, -37 µC and +3 µC, are fixed in place and separated by 2.8 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.)
how many m from the positive charge, not between the two charges?
To find the spot along the line where the net electric field is zero, we can use the principle of superposition. According to this principle, the total electric field at any point is the vector sum of the electric fields due to each individual charge.
Let's assume the positive charge (+3 µC) is located at the origin (0 m), and the negative charge (-37 µC) is at a point 2.8 m away. We need to find the position along the line where the net electric field is zero, relative to the positive charge.
To do this, we'll consider two cases:
1. Case 1: Net electric field due to the positive charge cancels out the electric field due to the negative charge.
2. Case 2: Net electric field due to the negative charge cancels out the electric field due to the positive charge.
For Case 1, we need to find the distance at which the electric field due to the positive charge (+3 µC) cancels out the electric field due to the negative charge (-37 µC).
Knowing the formula for electric field due to a point charge:
E = k * q / r^2
Where:
- E is the electric field
- k is the Coulomb's constant (9 x 10^9 Nm^2/C^2)
- q is the charge
- r is the distance between the charge and the point
Let's assume the distance from the positive charge to the spot is 'x' m. The electric field due to the positive charge at that point is given by:
E1 = k * (+3 µC) / x^2
The electric field due to the negative charge at that point is given by:
E2 = k * (-37 µC) / (2.8 m - x)^2
To find the spot where the electric fields cancel out, E1 and E2 must be equal:
E1 = E2
Substituting the values in the above equation, we get:
k * (+3 µC) / x^2 = k * (-37 µC) / (2.8 m - x)^2
Simplifying this equation, we can solve it to find the value of 'x'.
For Case 2, we would do a similar calculation but with the opposite charges:
E1 = k * (-37 µC) / x^2
E2 = k * (+3 µC) / (2.8 m - x)^2
Setting E1 = E2 and solving the equation will give us the distance for Case 2.
So, to summarize, there are two possible spots along the line where the net electric field is zero:
1. The distance between the positive charge and the spot for Case 1.
2. The distance between the positive charge and the spot for Case 2.
Using the above method, you can solve for 'x' in both cases to find the distances from the positive charge in meters.
To find the spot along the line where the net electric field is zero, we can use the concept of the superposition principle. The net electric field at any point due to the two charges is the vector sum of the electric fields created by each individual charge.
Let's assume that the positive charge (+3 µC) is located at the origin (0 m) and the negative charge (-37 µC) is located at the position (2.8 m).
The electric field due to a point charge at any point is given by the formula:
E = k * (Q / r^2)
Where:
E = electric field
k = electrostatic constant (approximated to 9 x 10^9 Nm^2/C^2)
Q = charge
r = distance from the charge to the point
The electric field created by the positive charge at any point along the line is always pointing away from the charge, since it is positive. On the other hand, the electric field created by the negative charge is always pointing towards the charge since it is negative.
So, to find the spot along the line where the net electric field is zero, we need to find a point where the magnitudes of the electric fields created by each charge are equal.
Let's solve this step-by-step:
Step 1: Calculate the electric field due to the positive charge at a point x meters away from it.
E_positive = (k * Q_positive) / (x)^2
Step 2: Calculate the electric field due to the negative charge at the point (2.8 m) - x meters away from it.
E_negative = (k * Q_negative) / (2.8 - x)^2
Step 3: Set the magnitudes of the electric fields equal to each other and solve for x.
(k * Q_positive) / (x)^2 = (k * Q_negative) / (2.8 - x)^2
Step 4: Simplify the equation.
Q_positive / x^2 = Q_negative / (2.8 - x)^2
Step 5: Cross multiply the equation.
Q_positive * (2.8 - x)^2 = Q_negative * x^2
Step 6: Expand and simplify the equation.
(2.8 - x)^2 = -37 µC / +3 µC * x^2
Step 7: Solve for x using the equation from Step 6.
(2.8 - x)^2 = -37/3 * x^2
2.8^2 - 5.6x + x^2 = -37/3 * x^2
7.84 - 5.6x + x^2 = (-37/3) * x^2
Multiply through by 3:
23.52 - 16.8x + 3x^2 = -37x^2
Combine like terms:
20.52 - 16.8x = -40x^2
Rearrange the equation:
40x^2 - 16.8x - 20.52 = 0
Step 8: Solve the quadratic equation using any suitable method (factoring, quadratic formula, etc.). In this case, the quadratic equation is not easily factorable, so we will use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
where a = 40, b = -16.8, and c = -20.52
Substituting the values:
x = (-(-16.8) ± √((-16.8)^2 - 4(40)(-20.52))) / (2(40))
x = (16.8 ± √(282.24 + 3260.8)) / 80
x = (16.8 ± √(3543.04)) / 80
x = (16.8 ± 59.55) / 80
Taking both positive and negative values of the square root, we get:
x1 = (16.8 + 59.55) / 80
x1 = 76.35 / 80
x1 = 0.954
x2 = (16.8 - 59.55) / 80
x2 = -42.75 / 80
x2 = -0.534
Therefore, the net electric field is zero at a spot 0.954 m away from the positive charge.
This question is pretty weird, but basically what you want to do is set these two charges up in an equation with the distance and have them equal to zero.
K(q_1)/(r+2.8)^2 - k(q_2)/r^2 = 0
q_1 - the first (negative) charge
q_2 - the second (positive) charge
r - this is the distance between the two charges where the net charge will equal zero
2.8 - this value will be different for other people I'm assuming, so it's just the original distance between the charges, so it basically stands for d.
At this point I'm hoping that you know how to solve for r using basic algebraic skills.
This is the right equation and I used it so it works. Also make sure you know that the spot is outside the two charges, not inside, which is why it's subtraction.
* As a side note, the K is Coulombs constant, however solving symbolically you will find that these will cancel
My final equation after all of this looked something like this:
sqrt(q_1/q_2) = 1 + d/r