To solve these problems, we need to understand the relationship between the electric field [tex]\(E\)[/tex] and the magnetic field [tex]\(B\)[/tex] in an electromagnetic wave. The speed of light [tex]\(c\)[/tex] connects these two fields, and it is given by [tex]\(c = 3 \times 10^8 \, m/s\)[/tex].
1. Calculating the maximum strength of the [tex]\(B\)[/tex]-field [tex]\(B_{max}\)[/tex] given the maximum [tex]\(E\)[/tex]-field [tex]\(E_{max} = 500 \, V/m\)[/tex]:
The relationship between [tex]\(E\)[/tex]-field and [tex]\(B\)[/tex]-field is given by:
[tex]\[
B_{max} = \frac{E_{max}}{c}
\][/tex]
Substituting the given values:
[tex]\[
B_{max} = \frac{500 \, V/m}{3 \times 10^8 \, m/s}
\][/tex]
Calculating this, we get:
[tex]\[
B_{max} = 1.67 \times 10^{-6} \, T
\][/tex]
So, the maximum strength of the [tex]\(B\)[/tex]-field is:
[tex]\[
B = 1.67 \times 10^{-6} \, T
\][/tex]
2. Calculating the maximum strength of the [tex]\(E\)[/tex]-field [tex]\(E_{max}\)[/tex] given the maximum [tex]\(B\)[/tex]-field [tex]\(B_{max} = 2.75 \times 10^{-6} \, T\)[/tex]:
Using the relationship between the fields:
[tex]\[
E_{max} = B_{max} \times c
\][/tex]
Substituting the given values:
[tex]\[
E_{max} = 2.75 \times 10^{-6} \, T \times 3 \times 10^8 \, m/s
\][/tex]
Calculating this, we get:
[tex]\[
E_{max} = 825 \, V/m
\][/tex]
So, the maximum strength of the [tex]\(E\)[/tex]-field is:
[tex]\[
E = 825 \, V/m
\][/tex]
To summarize:
1. The maximum strength of the [tex]\(B\)[/tex]-field when [tex]\(E_{max} = 500\, V/m\)[/tex] is:
[tex]\[
B = 1.67 \times 10^{-6} \, T
\][/tex]
2. The maximum strength of the [tex]\(E\)[/tex]-field when [tex]\(B_{max} = 2.75 \times 10^{-6} \, T\)[/tex] is:
[tex]\[
E = 825 \, V/m
\][/tex]
