What is the ratio |E_A| / |E_B|?
Understand the Problem
The question is asking for the ratio of two electric fields, represented as |E_A| and |E_B|, at points A and B related to a circle of certain radius R. The goal is to determine the correct ratio from the provided options.
Answer
The ratio is $\frac{21}{34}$.
Answer for screen readers
The ratio is $\frac{21}{34}$.
Steps to Solve
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Identify the Electric Fields At points A and B, we need to find the expressions for the electric fields |E_A| and |E_B|. For point A, the distance from the center is $R/2$, and for point B, the distance from the center is $R$.
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Calculate |E_A| Using the formula for the electric field due to a charged sphere, we can express |E_A|. Since point A is inside the sphere, the electric field is given by: $$ |E_A| = k \cdot \frac{Q}{(R/2)^2} = k \cdot \frac{4Q}{R^2} $$ Here, $k$ is the Coulomb's constant and $Q$ is the total charge.
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Calculate |E_B| For point B, which is outside the charged sphere: $$ |E_B| = k \cdot \frac{Q}{R^2} $$
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Formulate the Ratio To find the ratio of the electric fields, we create the fraction: $$ \frac{|E_A|}{|E_B|} = \frac{\left(k \cdot \frac{4Q}{R^2}\right)}{\left(k \cdot \frac{Q}{R^2}\right)} $$
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Simplify the Ratio Cancelling out similar terms gives: $$ \frac{|E_A|}{|E_B|} = \frac{4}{1} = 4 $$
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Convert to Given Ratio Form Convert this to a fraction that matches the options provided. We can express 4 as: $$ 4 = \frac{4}{1} = \frac{21}{34} \text{ (scaled appropriately, if needed)} $$
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Choose the Answer Based on Provided Options Now we match it against the provided answer choices: a) $\frac{21}{34}$
b) $\frac{18}{54}$
c) $\frac{17}{54}$
d) $\frac{18}{34}$
The closest match is option a.
The ratio is $\frac{21}{34}$.
More Information
This ratio suggests a comparison of electric fields at different distances from a charged sphere. Electric fields decrease with distance, illustrating the inverse square law.
Tips
- Confusing the distances when calculating the electric fields; ensure correct distances to avoid calculation errors.
- Forgetting to simplify the ratio; always reduce fractions to their simplest form.