Question

An aluminium wire of diameter 0.24 cm is connected in series to a copper wire of diameter 0.16 cm. The wires carry an electric current of 10 A. Determine the current density in aluminium wire.

Answer 1

Hint: We will first use $A = \pi {r^2}$ to find the area of the cross section of the aluminium wire. Then we will substitute it in the formula, $J = \dfrac{I}{A}$ to find the current density.

Complete step by step answer:
We know that, the diameter of aluminium wire: 0.24 cm
Therefore, radius of the aluminium wire: $r = \dfrac{d}{2}$
$ \Rightarrow r = \dfrac{{0.24}}{2}$cm
$ \Rightarrow r = 0.12$cm (convert it into meters)
$ \Rightarrow r = 0.12 \times {10^{ - 2}}$m
Now, area of cross section of the wire, $A = \pi {r^2}$
$ \Rightarrow A = 3.14 \times {\left( {0.12 \times {{10}^{ - 2}}} \right)^2}$
$ \Rightarrow A = 4.5 \times {10^{ - 6}}{m^2}$
So, from the formula of current density, we know that,
$J = \dfrac{I}{A}$ (where J is the current density; I is current; A is the cross-sectional area of the wire)
$ \Rightarrow J = \dfrac{{10}}{{4.5 \times {{10}^{ - 6}}}}$ (the value of I is given in the question and A we have already calculated above)
$ \Rightarrow J = 2.04 \times {10^6}A/{m^2}$
Additional information:
Current density is proportional to an electric field, and the direction of current density is the same as that of the electric field.
-If the electric field is uniform (i.e., constant) current density will be constant.
-If the electric field is zero (as in electrostatics inside a conductor), current density and hence current will be zero.
-If the current has not reached a steady-state, i.e., the flow of charge is not constant, then the current through different cross-sections at a particular instance may have different values.
-The direction of current density is the same as that of the velocity of positive charge or opposite to the direction of the velocity of negative charge.

Note: Current density is calculated differently for various situations, for example:
The current density at point P is given by $J = \dfrac{{di}}{{dA}}$. The current density when the current is not perpendicular to the area is given by, $J = \dfrac{{\Delta i}}{{\Delta A\cos \theta }}$. And, the current density in the case of conductors is given by, $J = \dfrac{I}{A}$. While electric current is a scalar quantity, the electric current density is a vector quantity.