A positive charge +q1 is located to the left of a negative charge -q2. On a line passing though the two charges, there are two places where the total potential is zero. The first place is between the charges and is 4.03 cm to the left of the negative charge. The second place is 7.01 cm to the right of the negative charge. (a) What is the distance between the charges? (b) Find q1/q2, the ratio of the magnitudes of the charges.
x1=4.03 cm, x2=7.01 cm.
d=?
The left point is A, the right point is B.
Potential due to the point charge is
φ = k•q/r.
Then, at the point A
φ(A1) =k•q1/(d-x1),
φ(A2) = -k•q2/x1,
φ(A1)+ φ(A2)=0 =>
k•q1/(d-x1) =k•q2/x1,
q1/q2=(d-x1)/x1….. (1)
At the point B
φ(B1) =k•q1/(d+x2),
φ(B2) = -k•q2/x2,
φ(B1)+ φ(B2)=0 =>
k•q1/(d+x2) =k•q2/x2,
q1/q2=(d+x2)/x2 …..(2)
Equate the right sides of the equations (1) and(2)
(d-x1)/x1 = (d+x2)/x2
d=2•x1•x2/(x2-x1) =
=24.03•7.01/(7.01-4.03) =18.96 cm.
q1/q2 =(d-x1)/x1 =
=(18.96-4.03)/4.03 = 3.7
To solve this problem, we'll use the concept of electric potential and apply it to the given situation.
First, let's consider the point between the charges where the potential is zero. We'll denote this point as P1. According to the problem, P1 is 4.03 cm to the left of the negative charge (-q2).
To find the distance between the charges, let's call it d, we can use the fact that the potential due to a point charge is given by the equation:
V = k * |q1| / r1 + k * |q2| / r2
Where:
- V is the electric potential
- k is the Coulomb's constant (k = 8.99 × 10^9 N m^2/C^2)
- q1 and q2 are the charges
- r1 and r2 are the distances from the charges to the point where the potential is measured
Since the potential at P1 is zero, the equation becomes:
0 = k * |q1| / r1 + k * |q2| / r2
Now, substitute the values given in the problem. We know that r1 = 4.03 cm and r2 = d - 7.01 cm (from the second point where the potential is zero).
The equation now becomes:
0 = k * |q1| / (4.03 cm) + k * |q2| / (d - 7.01 cm)
Now, let's move on to part (b) which is to find q1/q2, the ratio of the magnitudes of the charges.
Using the equation above, we can solve for q1/q2. Rearrange the equation to isolate q1/q2:
|q1| / |q2| = - (d - 7.01 cm) / 4.03 cm
Since we are only interested in the magnitudes of the charges, the negative sign can be ignored.
Now we can solve both equations for d (distance between the charges) and q1/q2 (ratio of the magnitudes of the charges).