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 a given distance from a long, straight filament with a known charge per unit length, you can use Gauss's law for electric fields, which states that the electric field at a distance r from an infinitely long charged filament is given by:
E = λ / (2πε₀r)
Where:
E is the electric field,
λ is the charge per unit length (in C/m),
ε₀ is the permittivity of free space (8.85 x 10^-12 C²/(N⋅m²)),
r is the distance from the filament (in meters).
In this case, we need to convert the charge per unit length from microcoulombs per meter (µC/m) to coulombs per meter (C/m) by dividing it by 10^6. Let's solve each part of the question.
(a) 10.0 cm from the filament:
First, convert 10.0 cm to meters: 10.0 cm = 0.10 m
Substituting the given values into the formula:
E = (-87.0 x 10^-6 C/m) / (2π(8.85 x 10^-12 C²/(N⋅m²))(0.10 m))
E = -0.985 N/C
So the electric field 10.0 cm from the filament is -0.985 N/C or -0.985 MN/C in millinewtons per coulomb.
(b) 26.0 cm from the filament:
Convert 26.0 cm to meters: 26.0 cm = 0.26 m
E = (-87.0 x 10^-6 C/m) / (2π(8.85 x 10^-12 C²/(N⋅m²))(0.26 m))
E = -0.378 N/C
The electric field 26.0 cm from the filament is -0.378 N/C or -0.378 MN/C.
(c) 140 cm from the filament:
Convert 140 cm to meters: 140 cm = 1.40 m
E = (-87.0 x 10^-6 C/m) / (2π(8.85 x 10^-12 C²/(N⋅m²))(1.40 m))
E = -0.0313 N/C
The electric field 140 cm from the filament is -0.0313 N/C or -0.0313 MN/C.