Physics for Scientists and Engineers: A Strategic Approach with Modern Physics (3rd Edition)

Published by Pearson
ISBN 10: 0321740904
ISBN 13: 978-0-32174-090-8

Chapter 30 - Current and Resistance - Conceptual Questions - Page 886: 9

Answer

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Work Step by Step

$$\color{blue}{\bf [a]}$$ According to the charge conservation principle, and since the current is given by $I=Q/t$, we can call it the conservation of the current principle, the current remains constant in the same wire. Thus, $$\boxed{I_1=I_2}$$ $$\color{blue}{\bf [b]}$$ The current density is given by $$J=\dfrac{I}{A}$$ where $A$ is the cross-sectional area of the wire. Hence, $$\dfrac{J_1}{J_2}=\dfrac{I_1/A_1}{I_2/A_2}$$ where $I_1=I_2$, $$\dfrac{J_1}{J_2}=\dfrac{A_2}{A_1}$$ and since $A_2\gt A_1$, $$\boxed{J_1\gt J_2}$$ $$\color{blue}{\bf [c]}$$ We know that the electric field is given by $$E=\dfrac{J}{\sigma}$$ where $\sigma$ is the conductivity Hence, $$\dfrac{E_1}{E_2}=\dfrac{J_1/\sigma_1}{J_2/\sigma_2}$$ where $\sigma_1=\sigma_2$ since the two parts are made of the same metal. $$\dfrac{E_1}{E_2}=\dfrac{J_1 }{J_2 }$$ and since $J_1\gt J_2$, $$\boxed{E_1\gt E_2}$$ $$\color{blue}{\bf [d]}$$ We know that current density is also given by $$J =nev_{\rm d}$$ where $n$ is the electron density and it is constant for the same metal. Hence, $$v_{\rm d}=\dfrac{J}{ne}$$ $$\dfrac{(v_{\rm d})_1}{(v_{\rm d})_2}=\dfrac{J_1/ne}{J_2/ne}$$ $$\dfrac{(v_{\rm d})_1}{(v_{\rm d})_2}=\dfrac{J_1 }{J_2 }$$ and since $J_1\gt J_2$, $$\boxed{{(v_{\rm d})_1}\gt {(v_{\rm d})_2}}$$
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