a point particle with charge q= 4/5 uC is placed on the x-axis at x=-10cm. A second particle of charge Q is now placed on the x-axis at x = 25cm, and its found that the electric field at the origin is zero. Find Q.
To find the charge Q, we will use the concept of electric field due to a point charge.
The electric field at a point due to a point charge can be calculated using the formula:
E = k * (Q / r^2)
Where:
E is the electric field,
k is the electrostatic constant (k = 9 × 10^9 Nm^2/C^2),
Q is the charge of the point charge, and
r is the distance between the point charge and the point where the electric field is being measured.
In this case, we know that the electric field at the origin is zero, which means that the electric field due to the second particle at the origin must be canceled out by the electric field due to the first particle at the origin.
So, the electric field E1 at the origin due to the first particle can be calculated as:
E1 = k * (q / r1^2)
Where:
E1 is the electric field due to the first particle,
q is the charge of the first particle (q = 4/5 μC),
and r1 is the distance between the first particle and the origin (r1 = 10 cm = 0.1 m).
Since we know that the electric field at the origin is zero, we can set E1 to zero:
0 = k * (4/5 μC) / (0.1 m)^2
Now, let's solve for the charge Q of the second particle:
0 = (9 × 10^9 Nm^2/C^2) * (4/5 μC) / (0.1 m)^2
Multiplying both sides by (0.1 m)^2:
0 = (9 × 10^9 Nm^2/C^2) * (4/5 μC)
Now, solve for Q:
Q = (0.1 m)^2 * (5/4 μC) / (9 × 10^9 Nm^2/C^2)
Simplifying the equation:
Q = (0.01 m^2) * (5/4 μC) / (9 × 10^9 Nm^2/C^2)
Q = (0.01 m^2) * (5/4 μC) / (9 × 10^9 Nm^2/C^2)
Q = (0.01 m^2) * (5/4) * (10^-6 C) / (9 × 10^9 Nm^2/C^2)
Q = (0.01) * (5/4) * (10^-6) / (9 × 10^9)
Q ≈ 5.556 * 10^-16 C
Hence, the charge Q of the second particle is approximately 5.556 * 10^-16 C.
To find the value of Q, we can use the principle of superposition. According to this principle, the total electric field at a point produced by multiple charges is the vector sum of individual electric fields created by each charge. In this case, we have two charges, q and Q.
We are given that the electric field at the origin (x = 0 cm) is zero. Let's consider the electric field at the origin due to each charge separately.
1. Electric Field due to q:
The electric field produced by a point charge q at a distance r from the charge is given by Coulomb's law:
E = k * q / r^2
where k is the Coulomb constant (k = 9 × 10^9 N m^2/C^2) and r is the distance from the charge to the point where we want to determine the electric field.
In this case, the distance from q to the origin is 10 cm = 0.1 m.
So, the electric field due to q at the origin is:
E1 = k * q / r^2 = (9 × 10^9 N m^2/C^2) * (4/5 × 10^-6 C) / (0.1 m)^2
2. Electric Field due to Q:
Similarly, let's calculate the electric field due to Q at the origin.
The distance from Q to the origin is 25 cm = 0.25 m.
So, the electric field due to Q at the origin is:
E2 = k * Q / r^2 = (9 × 10^9 N m^2/C^2) * Q / (0.25 m)^2
Since the total electric field at the origin is zero, the sum of the electric fields due to q and Q should be zero:
E1 + E2 = 0
Substituting the values, we get:
(9 × 10^9 N m^2/C^2) * (4/5 × 10^-6 C) / (0.1 m)^2 + (9 × 10^9 N m^2/C^2) * Q / (0.25 m)^2 = 0
Now, we can solve this equation to find the value of Q.