Question

An infinite line charge produces a field of \[7.182 \times {10^8}N/C\] at a distance of \[2cm\]. The linear charge density is:
A. \[7.27 \times {10^{ - 4}}C/m\]
B. \[7.98 \times {10^{ - 4}}C/m\]
C. \[7.11 \times {10^{ - 4}}C/m\]
D. \[7.04 \times {10^{ - 4}}C/m\]

Answer 1

Hint: The electric field is a physical property that is present around every electric charge. For a positive charge, the electric field line originates and comes out in search of negative charges. If there is no negative charge found then the field lines will extend to infinity. To answer the above question we have a relation that connects the electric field produced by the infinite line charges and the linear charge density. We need to substitute the values in this relation to get our answer.

Complete step by step solution:
Given that the electric field produced by the infinite line charge \[7.182 \times {10^8}N/C\]
The distance covered by the field lines, \[d = 2{\text{ }}cm\]
The formula given for the electric field produced by an infinite line charge is,
\[E = \dfrac{\lambda }{{2\pi { \in _0}d}}\]
Here \[E\] is the electric field produced
\[\lambda \] is the linear charge density.
\[{ \in _0}\] is the permittivity of free space
\[d\] is the distance
We know the value for \[\dfrac{1}{{4\pi { \in _0}}} = 9 \times {10^9}N{m^2}{C^{ - 2}}\]
Rearranging this, \[\dfrac{1}{{4 \times 9 \times {{10}^9}}} = \pi { \in _0}\]
Therefore substituting all the known values in the electric field equation we get,
\[\lambda = E \times 2\pi { \in _0}d\]
\[\lambda = 7.182 \times {10^8} \times 2 \times \dfrac{1}{{4 \times 9 \times {{10}^9}}} \times 0.02\] (Since \[2cm = 0.02m\] )
Solving the above equation we get,
\[\lambda = 7.98 \times {10^{ - 4}}C/m\]
Correct Answer:
The linear charge density is found to be \[7.98 \times {10^{ - 4}}C/m\].
Therefore the correct option is B.

Note:
The measure of the number of electric charges accumulated per unit area of a surface or unit volume of a body or a field can be described as charge density. Charge density is of three types based on area, volume, or length. The three types of charge density are linear charge density, surface charge density, and volume charge density. If we consider a point on a linear line of charge the measure of the charge distributed per unit length is described as the linear charge density. It can be represented by \[\lambda \].