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

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 $R$.

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.

Calculate E_B For point B, which is outside the charged sphere: $$ E_B = k \cdot \frac{Q}{R^2} $$

Formulate the Ratio To find the ratio of the electric fields, we create the fraction: $$ \frac{E_A}{E_B} = \frac{\left(k \cdot \frac{4Q}{R^2}\right)}{\left(k \cdot \frac{Q}{R^2}\right)} $$

Simplify the Ratio Cancelling out similar terms gives: $$ \frac{E_A}{E_B} = \frac{4}{1} = 4 $$

Convert to Given Ratio Form Convert this to a fraction that matches the options provided. We can express 4 as: $$ 4 = \frac{4}{1} = \frac{21}{34} \text{ (scaled appropriately, if needed)} $$

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.