Question

A very long straight wire has charge per unit length 1.44×10^-10C/m.

question:

At what distance from the wire is the magnitude of the electric field equal to 2.45 N/C ?

Use 8.85×10⁻¹²C²/(N*m²) for the permittivity of free space, and use π=3.14 .

Answer 1

Concepts and reason

The concept used here is the electric field due to straight wire.

The distance from the wire is calculated by using the relation between the electric field, distance and charge per unit length.

Fundamentals

Electric field is defined as the region in which an electric charge experiences electric force.

The electric field is given by,

$EA = \frac{q}{{{\varepsilon _0}}}$

Here, A is the area, q is the charge, and ${\varepsilon _0}$ is the permittivity.

The electric field due to a point charge is,

$EA = \frac{q}{{{\varepsilon _0}}}$

The charge is distributed uniformly. So, the charge on the wire is,

$q = \lambda l$

Here, $\lambda$ is the charge per unit length and l is the length of the wire.

Substitute $2\pi rl$ for A and $\lambda l$ for q.

$\begin{array}{c}\\E\left( {2\pi rl} \right) = \frac{{\lambda l}}{{{\varepsilon _0}}}\\\\E = \frac{\lambda }{{2\pi {\varepsilon _0}r}}\\\end{array}$

Calculate the distance from the wire.

Electric field due to the long straight wire is,

$E = \frac{\lambda }{{2\pi {\varepsilon _0}r}}$

Here, r is the distance from the wire, λ is the charge per unit length, and ${\varepsilon _0}$ is the permittivity of free space.

Rearrange the equation for r.

$r = \frac{\lambda }{{2\pi {\varepsilon _0}E}}$

Substitute 2.45 N/C for E, $1.44 \times {10^{ - 10}}{\rm{ C/m}}$ for λ, and $8.85 \times {10^{ - 12}}{\rm{ }}{{\rm{C}}^{\rm{2}}}{\rm{/N}} \cdot {{\rm{m}}^{\rm{2}}}$ for ${\varepsilon _0}$ .

$\begin{array}{c}\\r = \frac{{1.44 \times {{10}^{ - 10}}{\rm{ C/m}}}}{{2\left( {3.14} \right)\left( {8.85 \times {{10}^{ - 12}}{\rm{ }}{{\rm{C}}^{\rm{2}}}{\rm{/N}} \cdot {{\rm{m}}^{\rm{2}}}} \right)\left( {2.45{\rm{ N/C}}} \right)}}\\\\ = 1.06{\rm{ m}}\\\end{array}$

Ans:

At a distance of 1.06 m the magnitude of electric field is 2.45 N/C.