Two small beads having charges q1 and q2 are fixed so that the centers of the two beads are separated by a distance d shown in the figure. A third small bead of charge q=10.0 x 10-9 C is placed between both charges a distance x from q1.
a) If q1 = 31.0 x 10-9 C, q2 = 26.0 x 10-9 C, d = 0.35 m and x =1/3 d, what force is exerted on q due to both charges? Answer in units of Newtons and use the direction from q1 to q2 as the positive direction.
b) What force is exerted on q2 due to both charges q and q1?Answer in units of Newtons and use the direction from q1 to q2 as the positive direction.
c) If q1 = 31.0 x 10-9 C, q2 = - 26.0 x 10-9 C, d = 0.35 m and x =1/3 d, what force is exerted on q due to both charges? Answer in units of Newtons and use the direction from q1 to q2 as the positive direction.
To solve this problem, we can use Coulomb's law, which states that the force between two charged objects is directly proportional to the product of their charges and inversely proportional to the square of the distance between them.
a) The force exerted on q due to both charges (q1 and q2) can be determined by calculating the individual forces and adding them together.
1. Calculate the force exerted by q1 on q:
F1 = k * (q1 * q) / (x^2)
where k is the Coulomb's constant (k = 8.99 x 10^9 N m^2/C^2).
2. Calculate the force exerted by q2 on q:
F2 = k * (q2 * q) / ((d - x)^2)
3. Add the two forces to get the total force exerted on q:
F_total = F1 + F2
Now, let's plug in the given values and calculate the answer:
q1 = 31.0 x 10^(-9) C,
q2 = 26.0 x 10^(-9) C,
d = 0.35 m,
x = (1/3) * d.
Substitute these values into the formulas and calculate F1 and F2:
F1 = (8.99 x 10^9 N m^2/C^2) * (31.0 x 10^(-9) C) * (10.0 x 10^(-9) C) / [(1/3)^2 * (0.35)^2]
F2 = (8.99 x 10^9 N m^2/C^2) * (26.0 x 10^(-9) C) * (10.0 x 10^(-9) C) / [(2/3)^2 * (0.35)^2]
Finally, add F1 and F2 to get F_total, which will give you the force exerted on q due to both charges.
b) To find the force exerted on q2 due to both charges q and q1, you can follow a similar process:
1. Calculate the force exerted by q1 on q2:
F3 = k * (q1 * q2) / ((d - x)^2)
2. Calculate the force exerted by q on q2:
F4 = k * (q * q2) / (x^2)
3. Add the two forces to get the total force exerted on q2:
F_total_q2 = F3 + F4
Substitute the given values and calculate F3 and F4 to get the result.
c) For the case where q2 is negative, the force will still be calculated using Coulomb's law, but the direction is opposite. The positive direction is from q1 to q2, so if q2 is negative, the force will act in the opposite direction compared to the previous cases.
In this case, follow the same steps as before, but when calculating F2, use the negative value for q2 in the formula. The total force exerted on q due to both charges will again be F_total = F1 + F2.
a) To find the force exerted on q due to both charges, we need to calculate the force separately due to q1 and q2, and then add them up.
1. Calculate the force due to q1:
The force between q1 and q is given by Coulomb's law:
F1 = (k * |q1| * |q|) / r^2,
where k is the electrostatic constant (k = 9 × 10^9 N*m^2/C^2), |q1| and |q| are the magnitudes of q1 and q, and r is the distance between them.
Given:
q1 = 31.0 × 10^(-9) C,
q = 10.0 × 10^(-9) C,
r = x = 1/3d = (1/3) * 0.35 m = 0.1167 m.
Plugging in the values, we have:
F1 = (9 × 10^9 N*m^2/C^2) * (31.0 × 10^(-9) C) * (10.0 × 10^(-9) C) / (0.1167 m)^2.
2. Calculate the force due to q2:
The force between q2 and q is given by the same equation:
F2 = (k * |q2| * |q|) / r^2.
Given:
q2 = 26.0 × 10^(-9) C.
Plugging in the values, we have:
F2 = (9 × 10^9 N*m^2/C^2) * (26.0 × 10^(-9) C) * (10.0 × 10^(-9) C) / (0.1167 m)^2.
3. Add the forces together:
F_total = F1 + F2.
Calculate F1 and F2 separately using the given values, and then add them up to get the total force exerted on q due to both charges.
b) To find the force exerted on q2 due to both charges q and q1, we need to calculate the force separately due to q and q1, and then subtract q1's force from q's force.
1. Calculate the force due to q:
The force between q and q2 is given by Coulomb's law:
F_q = (k * |q| * |q2|) / r^2,
where |q| and |q2| are the magnitudes of q and q2, and r is the distance between them.
Given:
q = 10.0 × 10^(-9) C.
Plugging in the values, we have:
F_q = (9 × 10^9 N*m^2/C^2) * (10.0 × 10^(-9) C) * (26.0 × 10^(-9) C) / (0.1167 m)^2.
2. Calculate the force due to q1:
The force between q1 and q2 is given by the same equation:
F_q1 = (k * |q1| * |q2|) / r^2.
Given:
q1 = 31.0 × 10^(-9) C.
Plugging in the values, we have:
F_q1 = (9 × 10^9 N*m^2/C^2) * (31.0 × 10^(-9) C) * (26.0 × 10^(-9) C) / (0.1167 m)^2.
3. Subtract F_q1 from F_q:
F_total_q2 = F_q - F_q1.
Calculate F_q and F_q1 separately using the given values, and then subtract F_q1 from F_q to get the total force exerted on q2 due to both charges.
c) The steps to find the force on q due to both charges, when q2 is negative (-26.0 × 10^(-9) C), are the same as in part a). Follow the same steps outlined above, but use the value of q2 as negative in calculations.