A charge of 5.00 ✕ 10^-9 C and a charge of -3.50 ✕ 10^-9 C are separated by a distance of 60.0 cm. Find the position at which a third charge, of 14.0 ✕ 10^-9 C, can be placed so that the net electrostatic force on it is zero.
To find the position at which the net electrostatic force on the third charge is zero, we can use the principle of Coulomb's Law.
Coulomb's Law states that the electrostatic force between two charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. The formula is given by:
F = k * (|q1*q2| / r^2)
Where:
F is the electrostatic force,
k is the electrostatic constant (9 x 10^9 Nm^2/C^2),
q1 and q2 are the charges, and
r is the distance between the charges.
In this problem, we have two charges (5.00 x 10^-9 C and -3.50 x 10^-9 C) separated by a distance of 60.0 cm. We need to find the position at which a third charge of 14.0 x 10^-9 C can be placed so that the net electrostatic force on it is zero.
Let's assume the third charge is placed at a distance x from the positive charge and (60.0 - x) from the negative charge.
Using Coulomb's Law, we can equate the forces from the two charges to zero:
k * ((5.00 x 10^-9 C) * (14.0 x 10^-9 C) / x^2) = k * ((3.50 x 10^-9 C) * (14.0 x 10^-9 C) / (60.0 - x)^2)
Now, we can simplify and solve for x:
(5.00 x 10^-9 C * 14.0 x 10^-9 C) / x^2 = (3.50 x 10^-9 C * 14.0 x 10^-9 C) / (60.0 - x)^2
(70 x 10^-18 C^2) / x^2 = (49 x 10^-18 C^2) / (3600 - 120x + x^2)
Cross-multiply:
(70 x 10^-18 C^2) * (3600 - 120x + x^2) = (49 x 10^-18 C^2) * x^2
(252000 x 10^-18 C^2 - 8400 x^2 10^-18 C^2 + 70 x 10^-18 C^2 * x^2) = (49 x 10^-18 C^2) * x^2
Combine like terms:
(84000 x^2 10^-18 C^2) = 0
This equation has no solutions since the left-hand side is always positive, and the right-hand side is zero.
Therefore, there is no position at which the net electrostatic force on the third charge is zero with the given configuration of charges.
To find the position at which the net electrostatic force on the third charge is zero, we can use Coulomb's Law. Coulomb's Law states that the electrostatic force between two charges is directly proportional to the product of their magnitudes and inversely proportional to the square of the distance between them.
The formula for Coulomb's Law is:
F = k * |q1 * q2| / r^2
Where:
F is the electrostatic force between the charges,
k is the electrostatic constant (approximately equal to 9 * 10^9 N m^2/C^2),
q1 and q2 are the magnitudes of the two charges, and
r is the distance between the charges.
In this problem, we have three charges. Let's consider the charges as q1, q2, and q3.
Given:
q1 = 5.00 ✕ 10^-9 C (charge 1)
q2 = -3.50 ✕ 10^-9 C (charge 2)
q3 = 14.0 ✕ 10^-9 C (charge 3)
r12 = 60.0 cm (distance between charges 1 and 2)
r13 = r23 = unknown
Since we want the net electrostatic force on charge 3 to be zero, the forces between charge 3 and charges 1 and 2 must cancel each other out.
Using Coulomb's Law, we can calculate the individual forces between charge 3 and charges 1 and 2:
Force between charges 1 and 3:
F13 = k * |q1 * q3| / r13^2
Force between charges 2 and 3:
F23 = k * |q2 * q3| / r23^2
Since the net force should be zero, F13 + F23 = 0. Therefore, we can set the magnitudes of the forces equal to each other:
k * |q1 * q3| / r13^2 = k * |q2 * q3| / r23^2
We can cancel out the electrostatic constant, k, and solve for the ratio of r13^2 to r23^2:
|r1 * r3| / r13^2 = |r2 * r3| / r23^2
|r1 * r3| / |r2 * r3| = r13^2 / r23^2
|r1 * r3| / |r2 * r3| = r13^2 - r23^2
Taking the square root of both sides, we get:
|r1 * r3| / |r2 * r3| = √(r13^2 - r23^2)
|r1 / r2| = √(r13^2 - r23^2)
Simplifying further:
|r1 / r2| = r13 - r23
Notice that the charge magnitudes, q1, q2, and q3, do not affect the position at which the net force is zero. Only the distance ratios matter.
In this case, we have:
|r1 / r2| = 5.00 ✕ 10^-9 C / 3.50 ✕ 10^-9 C = 1.43
We can set up the equation again:
1.43 = r13 - r23
Since the distances, r13 and r23, are measured from the charges, we can set r23 = 0. We'll find r13:
1.43 = r13 - 0
Therefore, the position at which the net electrostatic force on the third charge is zero is at a distance of 1.43 meters from the first charge.
F = k Q/r^2
5 at 0 and -3.5 at 60
for 0
k (5)/x^2 + k(-3.5)/(x-60)^2=0
5(x-60)^2 - 3.5 x^2 = 0
5(x^2-120x+3600 ) - 3.5 x^2 = 0
1.5 x^2 - 600 x +18,000 = 0
x = 32.7 or 367
32.7 makes no sense (introduced by squaring)
so
367 cm