Certainly! Let's find the speed of light in material [tex]\( B \)[/tex] step-by-step.
### Step-by-Step Solution:
1. Understand the Relationship Between Refractive Index and Speed of Light:
The refractive index ([tex]\( n \)[/tex]) of a material is inversely proportional to the speed of light ([tex]\( v \)[/tex]) in that material. Mathematically, for two materials [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
[tex]\[
\frac{n_A}{n_B} = \frac{v_B}{v_A}
\][/tex]
Given:
[tex]\[
\frac{n_A}{n_B} = 1.41
\][/tex]
and the speed of light in material [tex]\( A \)[/tex] is:
[tex]\[
v_A = 1.05 \times 10^8 \text{ m/s}
\][/tex]
2. Express the Speed of Light in Material B Using the Given Ratios:
Rearrange the relationship to solve for [tex]\( v_B \)[/tex]:
[tex]\[
v_B = v_A \times \frac{n_B}{n_A}
\][/tex]
Since [tex]\( \frac{n_B}{n_A} = \frac{1}{1.41} \)[/tex], we get:
[tex]\[
v_B = v_A \times \frac{1}{1.41}
\][/tex]
3. Substitute the Given Values:
Plug the value of [tex]\( v_A \)[/tex] into the equation:
[tex]\[
v_B = 1.05 \times 10^8 \, \text{m/s} \times \frac{1}{1.41}
\][/tex]
4. Calculate the Result:
Perform the division:
[tex]\[
v_B = 1.05 \times 10^8 \, \text{m/s} \div 1.41
\][/tex]
Carrying out the division:
[tex]\[
v_B \approx 74468085.11 \, \text{m/s}
\][/tex]
### Conclusion:
The speed of light in material [tex]\( B \)[/tex] is approximately [tex]\( 7.45 \times 10^7 \, \text{m/s} \)[/tex] or [tex]\( 74468085.11 \, \text{m/s} \)[/tex].
Thus, the speed of light in material B is [tex]\( 74468085.11 \, \text{m/s} \)[/tex].
