four particles are fixed along an x axis, separated by distances d = 2.70 cm. The charges are q1 = +3e, q2 = -e, q3 = +e, and q4 = +6e, with e = 1.60 × 10-19 C. What is the value of the net electrostatic force on (a) particle 1 and (b) particle 2 due to the other particles?
F12= k•q1•q2/d²= k•3e²/d² (to the right)
F13= k•q1•q3/(2d)²= k•3e²/4d² (to the left)
F14= k•q1•q4/(3d)²= k•18e²/9d²(to the left)
F= k•3e²/d² - k•3e²/4d² - k•18e²/9d² = ...
F21= k•q1•q2/d²= k•3e²/d² (to the left)
F23= k•q2•q3/d²= k•e²/d² (to the right)
F24= k•q2•q4/(2d)²= k•6e²/4d²(to the right)
F= - k•3e²/d²+ k•e²/d²+ k•6e²/4d² =
...
To find the net electrostatic force on particle 1 and particle 2 due to the other particles, we can use Coulomb's law. Coulomb's law states that the magnitude of the electrostatic force between two charged particles is given by:
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 charges of the particles, and r is the distance between the particles.
Let's calculate the net electrostatic force on particle 1 first.
(a) Net electrostatic force on particle 1:
In this case, particle 1 is affected by the forces due to particle 2, particle 3, and particle 4.
The force on particle 1 due to particle 2 is given by:
F12 = k * (|q1| * |q2|) / (r12^2)
where r12 is the distance between particle 1 and particle 2. Since the particles are separated by distances d = 2.70 cm, r12 = d.
Substituting the given values into the equation, we get:
F12 = (8.99 x 10^9 N*m^2/C^2) * (|3e| * |-e|) / (2.70 cm)^2
The distance needs to be converted to meters, so 2.70 cm = 0.027 m.
F12 = (8.99 x 10^9 N*m^2/C^2) * (3e * e) / (0.027 m)^2
Now, calculate the magnitude of this force by plugging in the value for e:
F12 = (8.99 x 10^9 N*m^2/C^2) * (3 * 1.60 x 10^-19 C * 1.60 x 10^-19 C) / (0.027 m)^2
Calculate this expression to find the force exerted on particle 1 due to particle 2.
Repeat this process for the forces between particle 1 and particle 3, and particle 1 and particle 4. Then, sum up all three forces to find the net electrostatic force on particle 1.
(b) Similarly, you can calculate the net electrostatic force on particle 2 by considering the forces due to particle 1, particle 3, and particle 4.
Hope this helps!
To find the net electrostatic force on a particle due to other particles, we need to calculate the individual forces between the particles and then find the vector sum of these forces.
Let's start by finding the force on particle 1. The only particle that exerts a force on particle 1 is particle 2, since it is the closest neighbor.
To calculate the force between particle 1 and particle 2, we can use Coulomb's law:
F12 = (k * |q1 * q2|) / r^2
Where k is the electrostatic constant (k ≈ 9 × 10^9 Nm^2/C^2), r is the distance between the charges, and q1 and q2 are the charges on particle 1 and particle 2, respectively.
Plugging in the values:
k = 9 × 10^9 Nm^2/C^2
q1 = +3e = 3 * 1.60 × 10^-19 C
q2 = -e = -1 * 1.60 × 10^-19 C
r = d = 2.70 cm = 2.70 × 10^-2 m
F12 = (9 × 10^9 Nm^2/C^2) * |(3 * 1.60 × 10^-19 C) * (-1 * 1.60 × 10^-19 C)| / (2.70 × 10^-2 m)^2
Now, let's calculate this value:
F12 = (9 × 10^9 Nm^2/C^2) * (4.80 × 10^-19 C^2) / (7.29 × 10^-4 m^2)
= 0.622 N
So, the net electrostatic force on particle 1 due to particle 2 is 0.622 N in the opposite direction of particle 2.
Now, let's move on to finding the net electrostatic force on particle 2. This time, there are two particles exerting forces on particle 2, which are particle 1 and particle 3.
We can again use Coulomb's law to calculate the forces between particle 2 and particle 1 (F21) and between particle 2 and particle 3 (F23), and then find the vector sum:
F21 = (k * |q2 * q1|) / r^2
F23 = (k * |q2 * q3|) / r^2
Using the same values as before, we can calculate these forces:
F21 = (9 × 10^9 Nm^2/C^2) * (4.80 × 10^-19 C^2) / (7.29 × 10^-4 m^2) (force between particle 2 and particle 1)
F21 = 0.622 N (same as before)
F23 = (9 × 10^9 Nm^2/C^2) * (1.60 × 10^-19 C) * (1.60 × 10^-19 C) / (2.70 × 10^-2 m)^2 (force between particle 2 and particle 3)
F23 = 1.86 N
To find the net force on particle 2, we need to add the vector sum of F21 and F23:
Net force on particle 2 = F21 + F23
= 0.622 N + 1.86 N
= 2.48 N
So, the net electrostatic force on particle 2 due to the other particles is 2.48 N.