The charge per unit length on a long, straight filament is -87.0 µC/m.
(a) Find the electric field 10.0 cm from the filament. Distances are measured perpendicular to the length of the filament. (Take radially inward toward the filament as the positive direction.)
MN/C
(b) Find the electric field 26.0 cm from the filament.
MN/C
(c) Find the electric field 140 cm from the filament.
MN/C
To find the electric field at different distances from the filament, we can use the formula for the electric field created by a long straight filament:
E = λ / (2πε₀r)
where E is the electric field, λ is the charge per unit length on the filament, ε₀ is the permittivity of free space, and r is the distance from the filament.
Given: λ = -87.0 µC/m
(a) To find the electric field at a distance of 10.0 cm from the filament:
We need to convert the charge per unit length to Coulombs per meter:
λ = -87.0 µC/m = -87.0 × 10^(-6) C/m
Now we can substitute the given values into the formula:
E = (-87.0 × 10^(-6) C/m) / (2πε₀(0.1 m))
Using the value of ε₀ = 8.854 × 10^(-12) C²/(N·m²), we can calculate the electric field:
E = (-87.0 × 10^(-6) C/m) / (2π(8.854 × 10^(-12) C²/(N·m²))(0.1 m))
Calculate this expression to find the electric field at a distance of 10.0 cm from the filament.
(b) To find the electric field at a distance of 26.0 cm from the filament:
Use the same formula as above and substitute the given values:
E = (-87.0 × 10^(-6) C/m) / (2πε₀(0.26 m))
Again, calculate this expression to find the electric field at a distance of 26.0 cm from the filament.
(c) To find the electric field at a distance of 140 cm from the filament:
Use the same formula as above:
E = (-87.0 × 10^(-6) C/m) / (2πε₀(1.40 m))
Calculate this expression to find the electric field at a distance of 140 cm from the filament.
To find the electric field at different distances from the filament, we can use the concept of electric field due to an electric line charge.
The electric field (E) at a point perpendicular to the line charge filament is given by the expression:
E = (k * λ) / r
Where:
E is the electric field,
k is the electrostatic constant (k = 9.0 × 10^9 N * m^2 / C^2),
λ is the charge per unit length of the filament, and
r is the distance from the filament.
Now let's calculate the electric fields at the given distances:
(a) Distance = 10.0 cm = 0.1 m
λ = -87.0 µC/m = -8.7 × 10^-5 C/m
Substituting these values into the formula:
E = (9.0 × 10^9 N * m^2 / C^2) * (-8.7 × 10^-5 C/m) / 0.1 m
E ≈ -78.3 × 10^4 N/C = -7.83 MN/C (Negative sign indicates the direction is radially inward toward the filament)
Therefore, the electric field 10.0 cm from the filament is approximately -7.83 MN/C.
(b) Distance = 26.0 cm = 0.26 m
Substituting this value into the formula:
E = (9.0 × 10^9 N * m^2 / C^2) * (-8.7 × 10^-5 C/m) / 0.26 m
E ≈ -30.0 × 10^4 N/C = -3.0 MN/C
Therefore, the electric field 26.0 cm from the filament is approximately -3.0 MN/C.
(c) Distance = 140 cm = 1.40 m
Substituting this value into the formula:
E = (9.0 × 10^9 N * m^2 / C^2) * (-8.7 × 10^-5 C/m) / 1.40 m
E ≈ -0.56 × 10^4 N/C = -0.056 MN/C
Therefore, the electric field 140 cm from the filament is approximately -0.056 MN/C.