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Question #42826

Difficulty: 8/10 | 🟢 Free

The Question

A region of space contains a net electric field $\vec{E}$ composed of a static conservative component and a non-conservative induced component due to a localized time-varying magnetic field. In cylindrical coordinates $(r, \theta, z)$ , the net electric field in the $z=0$ plane is given by: $\vec{E} = E_0 \frac{a}{r} \hat{r} + E_0 \frac{r}{a} \hat{\theta}$ where $E_0$ is a positive constant with units of $\text{V}\cdot\text{m}^{-1}$ and $a$ is a positive constant with units of $\text{m}$ . A point charge $q$ (in Coulombs) is moved from point $P_1$ at $(r=a, \theta=0)$ to point $P_2$ at $(r=a e^{\pi}, \theta=\pi)$ along a logarithmic spiral path defined by $r(\theta) = a e^{\theta}$ . What is the total work done by the net electric field on the point charge $q$ during this motion?

Options

A.
$q E_0 a \left( \pi + \frac{e^{2\pi} - 1}{2} \right)$
B.
$q E_0 a \pi \left( 1 + e^{2\pi} \right)$
C.
$q E_0 a \pi$
D.
$q E_0 a \left( \pi + e^{\pi} - 1 \right)$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Intuition_Breaker

Train your brain to resist "obvious" answers that are designed to deceive your intuition.

Concepts Behind This Question

Electric Field Line Geometry|Potential vs Field Relationship|Dot Product Geometry

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to evaluate the work done by a field that includes a non-conservative component, explicitly challenging the assumption of path-independence.

A common trap for students is to treat the entire electric field as conservative (since static fields usually are) and calculate the work along a simpler, piecewise path (e.g., purely radial then purely azimuthal), which yields an incorrect result of $q E_0 a \pi (1 + e^{2\pi})$ .

Another trap is incorrectly writing the differential displacement vector in polar coordinates as $d\vec{l} = dr \hat{r} + a d\theta \hat{\theta}$ (using a constant initial radius for arc length), leading to the dimensionally consistent but numerically flawed Option D.

By forcing the integration along a specific logarithmic spiral $r(\theta)$ , the problem demands rigorous application of the vector dot product $\vec{E} \cdot d\vec{l}$ and parameterization of the path integral.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electric Charges and Fields - Class 12
Concept	Electric Field Line Geometry
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free