
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Consider a charged capacitor made with two square plates of side length  $L$ , uniformly charged, and separated by a very small distance  $d$ . The EMF across the capacitor is  $\xi$ . One of the plates is now rotated by a very small angle  $\theta$  to the original axis of the capacitor. Find an expression for the difference in charge between the two plates of the capacitor, in terms of (if necessary)  $d$ ,  $\theta$ ,  $\xi$ , and  $L$ . 
	
	Also, approximate your expression by transforming it to algebraic form: i.e. without any non-algebraic functions. For example, logarithms and trigonometric functions are considered non-algebraic. Assume  $ d << L $  and  $ \theta \approx 0 $ . 
	
	 $\emph{Hint}$ : You may assume that  $ \frac {\theta L}{d} $  is also very small. 

*Problem proposed by Trung Phan*

<details><summary>Clarification</summary>There are two possible ways to rotate the capacitor. Both were equally scored but this is what was meant: [asy]size(6cm);

real h = 7;
real w = 2;

draw((-w,0)--(-w,h));
draw((0,0)--(0,h), dashed);
draw((0,0)--h*dir(64));
draw(arc((0,0),2,64,90));
label(" $\theta$ ", 2*dir(77), dir(77));
[/asy]</details>

Present the answer in LaTex format: \boxed{Your answer}
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