Particles 1 and 2 of charge q1 = q2 = +3.20 x 10-19 C are on a y axis at distance d = 22.0 cm from the origin. Particle 3 of charge q3 = +9.60 x 10-19 C is moved gradually along the x axis from x = 0 to x = +5.0 m.
At what value of x will the magnitude of the electrostatic force on the third particle from the other two particles be maximum?
What is the maximum magnitude?
To find the value of x at which the magnitude of the electrostatic force on the third particle is maximum, we need to analyze the forces between the particles.
The electrostatic force between two charged particles is given by Coulomb's Law:
F = k * (|q1| * |q2|) / r^2
Where:
F is the electrostatic force,
k is the electrostatic constant (k = 8.99 x 10^9 N m^2/C^2),
q1 and q2 are the magnitudes of the charges,
|q1| and |q2| are the absolute values of the charges,
r is the distance between the charges.
In this case, the force on the third particle is caused by the two other particles. Let's calculate the force exerted on the third particle due to particle 1 and particle 2 individually.
Force from particle 1 on the third particle:
F1 = k * (|q1| * |q3|) / r1^2
Force from particle 2 on the third particle:
F2 = k * (|q2| * |q3|) / r2^2
Since the magnitudes of q1, q2, and q3 are all positive, the directions of the forces will depend on the positions of the particles. We need to consider the distance from each particle to the third particle.
Let's calculate the distances r1 and r2:
r1 = x - d (Distance from particle 1 to the third particle)
r2 = x + d (Distance from particle 2 to the third particle)
Next, substitute these values into the force equations:
F1 = k * (|q1| * |q3|) / (x - d)^2
F2 = k * (|q2| * |q3|) / (x + d)^2
To find the maximum of the magnitude of the force, we need to take the derivative of the total force Ftotal with respect to x and set it equal to zero:
Ftotal = F1 + F2
dFtotal/dx = dF1/dx + dF2/dx = 0
Differentiating F1 and F2 with respect to x:
dF1/dx = -2 * k * (|q1| * |q3|) / (x - d)^3
dF2/dx = 2 * k * (|q2| * |q3|) / (x + d)^3
Setting dF1/dx + dF2/dx = 0:
-2 * k * (|q1| * |q3|) / (x - d)^3 + 2 * k * (|q2| * |q3|) / (x + d)^3 = 0
Now solve this equation for x, which will give us the position at which the magnitude of the electrostatic force on the third particle is maximum.
After calculating x, we can substitute it back into either F1 or F2 to find the maximum magnitude of the electrostatic force.