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Question: Current density in a Cu wire is \(2.5\times {{10}^{8}}A{{m}^{-2}}\). If 8A current is flowing throug...

Current density in a Cu wire is 2.5×108Am−22.5\times {{10}^{8}}A{{m}^{-2}}. If 8A current is flowing through it, diameter of the wire is _ _ _ _ _
A. 0.2mm
B. 0.2cm
C. 0.2m
D. 2mm

Explanation

Solution

Current density is a physical quantity related to current electricity. We know that current is the rate of flow of charge with respect to time through a conductor. Current density is defined as the amount of current flowing through per unit area of a cross section.

Formula used:
J=iAJ=\dfrac{i}{A}
A=π(d2)2A=\pi {{\left( \dfrac{d}{2} \right)}^{2}}

Complete step by step answer:
Let us first understand what is meant by the quantity current density. Current density is defined as the amount of current flowing through per unit area of a cross section. Suppose we have a wire of some cross section and let a current be flowing through this wire, then if the current is uniform then the current density is equal to the ratio of the current in the wire to the area of the cross section of the wire. If the cross sectional area of the wire is A and the current flowing in it is i, then the current density is given as J=iAJ=\dfrac{i}{A} ….. (i)

In the given case, the cross section of the Cu wire in circular and the area of a circle is given to be A=π(d2)2A=\pi {{\left( \dfrac{d}{2} \right)}^{2}}, where d is the diameter of the cross section.It is given that current density in the wire is 2.5×108Am−22.5\times {{10}^{8}}A{{m}^{-2}} and the current flowing in it is 8A. This means that, J=2.5×108Am−2J=2.5\times {{10}^{8}}A{{m}^{-2}} and i=8Ai=8A.
Substitute the values of J, i and A in (i).
2.5×108=8π(d2)22.5\times {{10}^{8}}=\dfrac{8}{\pi {{\left( \dfrac{d}{2} \right)}^{2}}}.
⇒2.5×108=8×4πd2\Rightarrow 2.5\times {{10}^{8}}=\dfrac{8\times 4}{\pi {{d}^{2}}}
⇒d2=8×4π2.5×108 ⇒d2=4×10−8\Rightarrow {{d}^{2}}=\dfrac{8\times 4}{\pi 2.5\times {{10}^{8}}}\\\ \Rightarrow {{d}^{2}} =4\times {{10}^{-8}}
⇒d=4×10−8 ⇒d=2×10−4m ∴d=0.2mm\Rightarrow d=\sqrt{4\times {{10}^{-8}}}\\\ \Rightarrow d =2\times {{10}^{-4}}m\\\ \therefore d =0.2mm.
This means that the diameter of the wire is equal to 0.2 mm.

Hence, the correct option is A.

Note: Current density is considered to be a vector quantity. The direction of current density is taken along the direction of the area vector of the cross section.Area in some cases is considered a vector (called area vector). The area vector is along an axis normal to the area. In the case of current density, the direction of the area vector is in the direction of the flow of the current.