Two point charges 5Nc & -2NC are located at (2,0,4) & (-3,0,5) respectively.
(a) Determine the force on 1Nc point charge located at (1,-3,7).
(b) Also find E at (1,-3,7)
(a) Well, well, well, let's calculate the force! We have two point charges, one with 5Nc and the other with -2NC. They are located at (2,0,4) and (-3,0,5) respectively. And we are looking for the force on a 1Nc point charge at (1,-3,7).
To calculate the force, we need to use Coulomb's Law, which states that the force between two charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between them.
So, using Coulomb's Law, the force (F) between two charges is given by the equation:
F = k * (q1 * q2) / r^2
where k is the electrostatic constant, q1 and q2 are the charges, and r is the distance between them.
Now, let's plug in the values and get cracking! The electrostatic constant (k) is 9 × 10^9 Nm^2/C^2.
First, let's calculate the distance (r) between the point charges. It's the square root of the sum of the squares of the differences in their x, y, and z coordinates.
r = sqrt((2 - 1)^2 + (0 - (-3))^2 + (4 - 7)^2)
r = sqrt(1^2 + 3^2 + (-3)^2)
r = sqrt(1 + 9 + 9)
r = sqrt(19)
Now, let's calculate the force:
F = (9 × 10^9 Nm^2/C^2) * ((5 × 10^-9 C) * (1 × 10^-9 C)) / (sqrt(19))^2
F = (9 × 10^9 Nm^2/C^2) * (5 × 10^-9 C * 1 × 10^-9 C) / 19
F ≈ 2.368421 × 10^-9 N
Therefore, the force on the 1Nc point charge located at (1,-3,7) is approximately 2.368421 × 10^-9 Newtons.
(b) Now let's move on to finding the electric field (E) at the point (1,-3,7). The electric field is a vector quantity that describes the force per unit charge experienced by a test charge placed at a given point.
To calculate the electric field, we can use the equation:
E = F / q
where E is the electric field, F is the force, and q is the charge of the test charge.
Plugging in the values we've got, the force (F) is approximately 2.368421 × 10^-9 N and the charge (q) is 1Nc.
Therefore, the electric field (E) at (1,-3,7) is approximately 2.368421 × 10^-9 N/C.
Voila! We're done!
To determine the force on the 1 Nc point charge located at (1,-3,7), we can use Coulomb's Law, which states that the force between two point charges is given by the formula:
F = (k * |q1 * q2|) / r^2
where:
- F is the force between the charges,
- 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,
- r is the distance between the charges.
Let's calculate the force:
1. Calculate the distance between the two charges:
- Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2)
- Distance = sqrt((-3 - 1)^2 + (0 - (-3))^2 + (5 - 7)^2)
- Distance = sqrt((-4)^2 + (3)^2 + (-2)^2)
- Distance = sqrt(16 + 9 + 4)
- Distance = sqrt(29)
2. Calculate the force:
- F = (k * |q1 * q2|) / r^2
- F = (8.99 x 10^9 * |1 * (-2)|) / (sqrt(29))^2
- F = (8.99 x 10^9 * 2) / 29
- F ≈ 617,931.03 N
Therefore, the force on the 1 Nc point charge located at (1,-3,7) is approximately 617,931.03 N.
To find the electric field (E) at (1,-3,7), we can use the following formula:
E = (k * |q|) / r^2
where:
- E is the electric field,
- k is the electrostatic constant,
- q is the charge,
- r is the distance.
Let's calculate the electric field:
3. Calculate the distance between the first charge and the point where the electric field is to be found:
- Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2)
- Distance = sqrt((1 - 2)^2 + (-3 - 0)^2 + (7 - 4)^2)
- Distance = sqrt((-1)^2 + (-3)^2 + (3)^2)
- Distance = sqrt(1 + 9 + 9)
- Distance = sqrt(19)
4. Calculate the electric field:
- E = (k * |q|) / r^2
- E = (8.99 x 10^9 * 1) / (sqrt(19))^2
- E = (8.99 x 10^9) / 19
- E ≈ 473,157,894.74 N/C
Therefore, the electric field at (1,-3,7) is approximately 473,157,894.74 N/C.
To determine the force on the 1Nc point charge located at (1,-3,7) due to the other point charges, we can use Coulomb's Law. Coulomb's Law states that the force between two point charges is directly proportional to the product of their magnitudes and inversely proportional to the square of the distance between them.
(a) Force on 1Nc point charge:
1. Calculate the distance between the point charge at (1,-3,7) and the first point charge at (2,0,4):
- Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2)
- Distance = sqrt((2 - 1)^2 + (0 - (-3))^2 + (4 - 7)^2)
- Distance = sqrt(1^2 + 3^2 + (-3)^2)
- Distance = sqrt(1 + 9 + 9)
- Distance = sqrt(19)
2. Calculate the force between the two charges using Coulomb's Law:
- Force = (k * |q1 * q2|) / Distance^2
- where:
- k is the electrostatic constant (k = 8.99 * 10^9 Nm^2/C^2)
- q1 and q2 are the magnitudes of the charges
- Distance is the distance between the charges
- Force = (8.99 * 10^9 Nm^2/C^2) * (|1 Nc * 5 Nc|) / (sqrt(19))^2
- Force = (8.99 * 10^9 Nm^2/C^2) * (5 Nc) / 19
- Force = (8.99 * 10^9 Nm^2/C^2) * (5 Nc) / 19
- Force ≈ 2.33 N
Therefore, the force on the 1Nc point charge located at (1,-3,7) due to the other point charges (5Nc and -2NC) is approximately 2.33 N in magnitude.
(b) Electric field (E) at (1,-3,7):
The electric field at a point due to a charge is defined as the force experienced per unit positive charge placed at that point.
1. To find the electric field at (1,-3,7), we can again use Coulomb's Law and the formula for electric field:
- Electric field (E) = (k * |q|) / Distance^2
- where:
- k is the electrostatic constant (k = 8.99 * 10^9 Nm^2/C^2)
- q is the magnitude of the point charge
- Distance is the distance between the point charge and the point where we want to find the electric field
2. Calculate the electric field due to the first point charge at (2,0,4):
- Electric field = (8.99 * 10^9 Nm^2/C^2) * (5 Nc) / (sqrt(19))^2
- Electric field = (8.99 * 10^9 Nm^2/C^2) * (5 Nc) / 19
3. Calculate the electric field due to the second point charge at (-3,0,5):
- Electric field = (8.99 * 10^9 Nm^2/C^2) * (|-2 Nc|) / (sqrt(19))^2
- Electric field = (8.99 * 10^9 Nm^2/C^2) * (2 Nc) / 19
4. Add the electric field vectors from both charges at (1,-3,7) to find the net electric field at that point.
Therefore, to find the net electric field at (1,-3,7), calculate the electric fields at that point due to each of the two charges using Coulomb's Law and then sum the individual electric field vectors.