Suppose the charge q2 can be moved left or right along a line connecting charges q1 and q3. Given that q = +15 µC, find the distance from q1 where q2 experiences a net electrostatic force of zero? (The charges q1 and q3 are separated by a fixed distance of 36 cm.)
q1=+q, q2=-2.0q, q3=+3.0q
magnitude in cm ?
direction?
F(q2) = ke*q1*q2/(r12)^2 + ke*q3*q2/(r23)^2 = 0
or
q1*q2/(r12)^2 = q3*q2/(r23)^2
q*-2q/(r12)^2 = 3q*-2q/(r23)^2
-2*q^2/(r12)^2 = -6q^2/(r23)^2
or
1/(r12)^2 = 3/(r23)^2
where r12 is the distance between charge 1 and 2; r23 is the distance bewteen charge 2 and 3. For symmetry sake, put q2 and zero. Then q2 must be between q1 and q3, and (r12)^2 is the same as r1^2, when r1 is the coordinate of q1; r23^2 is the same as r3^2, where r3 is the coordinate of q3.
you know that -r1 + r3 = 36, or r3 = 36 + r1. Substitute this into the above equation and solve for r1, the distance from q1 where q2 experiences a net electrostatic force of zero
Why did the charge q2 cross the road? To find the distance from q1 where it experiences a net electrostatic force of zero! Let's calculate that, but first, let me ask you this: have you ever seen a charged clown doing calculations? It's electrifying! Now, back to business before I get too charged up.
To find the distance, we can use the principle that the net electrostatic force on q2 will be zero when the forces due to q1 and q3 cancel each other out. The magnitude of the electrostatic force between two charges can be calculated using Coulomb's Law, which states that F = k * (|q1|*|q2|) / r^2, where k is Coulomb's constant and r is the distance between the charges.
Given q1 = +15 µC, q2 = -2.0q, q3 = +3.0q, and the fixed distance between q1 and q3 is 36 cm, we can write the equation for the net electrostatic force on q2 as:
k * (|q1|*|q2|) / r1^2 - k * (|q2|*|q3|) / r2^2 = 0
Simplifying further, we have:
k * (|q1|*|q2|) / r1^2 = k * (|q2|*|q3|) / r2^2
Now, substitute the given values into the equation and solve for r1:
k * (|+15 µC| * |-2.0q|) / r1^2 = k * (|-2.0q| * |+3.0q|) / (36 cm)^2
Since the magnitudes of the charges are the same, we can cancel them out:
k * (15 µC * |-2.0|) / r1^2 = k * (|-2.0| * |+3.0|) / (36 cm)^2
The values for k and the charges are constants, so they can be eliminated:
(15 µC * 2.0) / r1^2 = 2.0 * 3.0 / (36 cm)^2
Now, let's solve for r1:
r1^2 = (15 µC * 2.0 * (36 cm)^2) / (2.0 * 3.0)
r1^2 = 6.75 µC * (36 cm / 3.0)
r1^2 = 6.75 µC * 12 cm
r1^2 = 81 µC * cm
Taking the square root of both sides:
r1 = √(81 µC * cm)
r1 ≈ 9 µC * cm
Thus, the distance from q1 where q2 experiences a net electrostatic force of zero is approximately 9 µC * cm. That's quite a shocking result, isn't it?
To find the distance from q1 where q2 experiences a net electrostatic force of zero, we need to consider the electrostatic forces exerted by q1 and q3 on q2. By applying Coulomb's Law, the magnitude of the electrostatic force between two charges is given by the equation:
F = k * |q1*q2| / r^2
Where:
F is the electrostatic force
k is the Coulomb's constant (8.99 x 10^9 N m^2/C^2)
q1 and q2 are the charges
r is the distance between the charges
In this case, q1 = +15 µC (or 15 x 10^-6 C)
q2 = -2.0q1 = -2.0 * 15 x 10^-6 C = -30 x 10^-6 C
q3 = +3.0q1 = 3.0 * 15 x 10^-6 C = 45 x 10^-6 C
r = 36 cm = 0.36 m
Since q2 experiences a net electrostatic force of zero, the forces exerted by q1 and q3 on q2 must be equal in magnitude but opposite in direction.
Considering the direction of the forces, the force between q1 and q2 will be attractive since q1 is positive and q2 is negative. The force between q2 and q3 will also be attractive since q2 and q3 are both negative.
Let's first calculate the magnitude of the forces:
F1 = k * |q1*q2| / r^2 = (8.99 x 10^9 N m^2/C^2) * (15 x 10^-6 C) * (30 x 10^-6 C) / (0.36 m)^2
F1 = 3.062 x 10^-8 N
F3 = k * |q2*q3| / r^2 = (8.99 x 10^9 N m^2/C^2) * (30 x 10^-6 C) * (45 x 10^-6 C) / (0.36 m)^2
F3 = 6.428 x 10^-8 N
Now, since the forces exerted by q1 and q3 on q2 are equal in magnitude but opposite in direction, we can set up the equation:
|F1| = |F3|
Taking the magnitudes of both forces:
|3.062 x 10^-8 N| = |6.428 x 10^-8 N|
Simplifying:
3.062 x 10^-8 N = 6.428 x 10^-8 N
Since this equation is not true, it means that the forces are not balanced. Therefore, there is no distance from q1 where q2 experiences a net electrostatic force of zero in this scenario.
To find the distance from q1 where q2 experiences a net electrostatic force of zero, we can start by analyzing the forces acting on q2.
The magnitude of the electrostatic force between two charges q1 and q2 is given by Coulomb's Law:
F = k * |q1| * |q2| / r^2
Where F is the force, k is Coulomb's constant (8.99 x 10^9 N m^2 / C^2), |q1| and |q2| are the magnitudes of the charges, and r is the distance between the charges.
In this case, q1 = +q, q2 = -2.0q, and q3 = +3.0q.
Now, let's consider q2 at a distance x from q1. The force between q1 and q2 is attractive, while the force between q2 and q3 is repulsive. Therefore, the net force on q2 can be found by subtracting the magnitudes of the forces between q1 and q2 and between q2 and q3:
F_net = F_q1q2 - F_q2q3
Since we want the distance where the net force is zero, we can set F_net = 0 and solve for x.
F_q1q2 = k * |q1| * |q2| / x^2
F_q2q3 = k * |q2| * |q3| / (36 - x)^2
Setting F_net = 0:
0 = F_q1q2 - F_q2q3
0 = k * |q1| * |q2| / x^2 - k * |q2| * |q3| / (36 - x)^2
Now, substitute the given values:
0 = k * (q) * (-2.0q) / x^2 - k * (-2.0q) * (3.0q) / (36 - x)^2
Simplifying the equation:
0 = (2q^2) / x^2 - (6q^2) / (36 - x)^2
To proceed, we can multiply both sides of the equation by x^2 * (36 - x)^2 to eliminate the denominators. This simplifies the equation to:
0 = 2q^2 * (36 - x)^2 - 6q^2 * x^2
Next, expand the squares:
0 = 2q^2 * (1296 - 72x + x^2) - 6q^2 * x^2
Simplifying further:
0 = 2592q^2 - 144q^2x + 2q^2x^2 - 6q^2x^2
Combining like terms:
0 = 2592q^2 - 144q^2x - 4q^2x^2
Now, we can factor out q^2 common to all terms:
0 = q^2 * (2592 - 144x - 4x^2)
Setting each factor equal to zero:
q^2 = 0 or 2592 - 144x - 4x^2 = 0
Since q cannot be zero (given as +15 µC), we can focus on the second factor:
2592 - 144x - 4x^2 = 0
Now, we can solve this quadratic equation to find the distance x at which q2 experiences a net electrostatic force of zero. The equation can be rewritten in standard quadratic form:
4x^2 + 144x - 2592 = 0
To solve for x, we can factor or use the quadratic formula:
Using the quadratic formula:
x = (-b ± sqrt(b^2 - 4ac)) / (2a)
In this case, a = 4, b = 144, and c = -2592. Plugging these values into the formula:
x = (-144 ± sqrt(144^2 - 4 * 4 * -2592)) / (2 * 4)
x = (-144 ± sqrt(20736 + 41568)) / 8
x = (-144 ± sqrt(62304)) / 8
x = (-144 ± 249.6) / 8
x = (-144 + 249.6) / 8 or x = (-144 - 249.6) / 8
x = 105.6 / 8 or x = -393.6 / 8
x = 13.2 cm or x = -49.2 cm
The distance from q1 where q2 experiences a net electrostatic force of zero is 13.2 cm. Since distance cannot be negative, the direction is towards q3.