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?
Between the charges, E is never zero, as it is in the same direction (towards the -17). To the left of the two charges, assuming the -17 is leftmost, one has E from the - charge to the right,and E from the + charge to the left. I suspect there wont be a zero there, lets check:
Etotal=17k/x^2-3.6k/(3.2+x)^2=0?
17(3.2^2+6.4x+x^2)=3.6x^2
I am working this now approxiamtely, in my head, you do it accurately.
173+106x+17x^2-3.6x^2=0
12.4x^2+106x+173)=0
x=(-106+-sqrt(106^2-4*12.4*173))/25
x=about -4+-2.5
but x has to be + in this model. No solution to the left.
Now to the right of the 3.6microC, same idea
E=3.6k/x^2-17k/(x+3.2)^2=0
this changes the equation to
17x^2=3.6(x^2+6.4x+10.2)
13.4x^2-20.5x-37=0 appx
x=(20.5+-sqrt(20.5^2+4*13.4*37)/26.8
= 20.5/26.8 +-48/26.8
= .76+- 1.79
but x must be positive, so x=2.55m to the right of the positive charge. Of course, check it, I did it in my head.
To find the spot along the line where the net electric field is zero, we can use the principle of superposition of electric fields. This principle states that the electric field at a point due to multiple charges is equal to the vector sum of the electric fields produced by each individual charge.
Let's denote the position of the negative charge (-17 µC) as Q1 and the position of the positive charge (+3.6 µC) as Q2. We need to find the position, which we'll call x, along the line where the net electric field is zero.
(a) To find x, we need to consider the electric fields produced by each charge. The electric field due to a point charge is given by Coulomb's Law:
E = k * Q / r^2
Where E is the electric field, k is Coulomb's constant (8.99 x 10^9 Nm^2/C^2), Q is the charge, and r is the distance between the charge and the point where the electric field is being measured.
Let's set up the equation for the net electric field at the spot x:
E1 + E2 = 0
k * Q1 / (x + d)^2 + k * Q2 / (d - x)^2 = 0
where d is the separation between the two charges (3.2 m).
Substituting the values, we get:
(8.99 x 10^9 Nm^2/C^2) * (-17 x 10^-6 C) / (x + 3.2)^2 + (8.99 x 10^9 Nm^2/C^2) * (3.6 x 10^-6 C) / (3.2 - x)^2 = 0
We can solve this equation for x.
(b) Once we find the position x where the net electric field is zero, we can calculate the force on a charge of +1 µC placed at this spot using Coulomb's Law:
F = k * |q1| |q2| / r^2
Where F is the force, k is Coulomb's constant, q1 and q2 are the charges, and r is the distance between the charges.
F = (8.99 x 10^9 Nm^2/C^2) * (1 x 10^-6 C) * (3.6 x 10^-6 C) / (x - x2)^2
To find the spot along the line through the charges where the net electric field is zero, you can use the concept of electric field superposition. The net electric field at a point in space is the vector sum of the electric field contributions from each individual charge.
(a) To find the spot where the net electric field is zero, we need to determine the positions of the charges relative to that spot. Let's assume that the positive charge (+3.6 µC) is located at the origin (0, 0) and the negative charge (-17 µC) is located at position (3.2 m, 0).
Now, we can analyze the electric field contributions from each charge separately and find the spot where they cancel out.
The electric field, E, from a point charge is given by the equation:
E = k * |q| / r^2
where k is the Coulomb's constant (8.99 x 10^9 Nm^2/C^2), |q| is the magnitude of the charge, and r is the distance from the charge to the spot where we want to calculate the field.
For the positive charge (+3.6 µC), the electric field at any point along the line would be directed away from it.
For the negative charge (-17 µC), the electric field at any point along the line would be directed towards it.
Considering the magnitudes of the charges, we can see that the electric field contribution from the positive charge will be weaker than the electric field contribution from the negative charge. Therefore, the spot where the net electric field is zero should be closer to the positive charge.
To determine the exact location, we need to calculate the electric field contributions from each charge at different points along the line, and add them up. The algebra for this process can be quite complex, so it is best to use numerical methods or suitable software to solve it.
(b) Once we have determined the spot where the net electric field is zero, we can find the force on a charge of +1 µC placed at that spot using the equation:
F = q * E
where F is the force, q is the charge, and E is the electric field at that spot.
Substituting the values, we can calculate the force on the charge. Remember, since the net electric field is zero at this spot, the force on the charge will also be zero.
It is important to note that these calculations require knowledge of vector addition and the principles of electric fields in physics.