Question

An electric current passes through a non-uniform cross-section wire made of homogenous and isotropic material. If the $j_A$ and $j_B$ be the current densitites and $E_A$ and $E_B$ be the electric field intensities at A and respectively, then

• A. $j_A$ < $j_B$,$E_A$ < $E_B$
• B. $j_A$ <$j_B$,$E_A$ > $E_B$
• C. $j_A$ > $j_B$,$E_A$ < $E_B$
• D. $j_A$ > $j_B$,$E_A$ > $E_B$

Answer 1

The correct option is D $j_A$ > $j_B$,$E_A$ > $E_B$

Step 1: Current in conductor

The current flowing is same at every cross section throughout the conductor.

If any charge enters through a cross section in any part of conductor an equal amount of charge must leave out through another cross section. This is because there is no accumulation or roduction of charge in any part of conductor, they only continuously flow through every cross section.

Step 2: Current density

We know, current density $j=\frac{I}{A}$

So, the current density is inversely proportional to the cross-sectional area of wire for constant current.

Hence , $j_A$ > $j_B$ ($as{A}_B$ < $A_A$)

Step 3: Electric field
From ohm's law, $J=\sigma E$
$J=\frac{E}{\delta }$ $(\delta \to Resistivity)$

Since, materials is same $\delta$ is constant

$\therefore E\propto J$

As $(J{)}_A$ > $(J{)}_B$,$\Rightarrow$$(E{)}_A$ > $(E{)}_B$

Hence, option (a) is correct.