To find the magnitude of the magnetic field at a different distance from the same wire, we use the relationship between the magnetic field [tex]\( B \)[/tex] and the distance [tex]\( r \)[/tex] from a long, straight current-carrying wire. The magnetic field is inversely proportional to the distance from the wire, which means:
[tex]\[ B \propto \frac{1}{r} \][/tex]
Given the initial conditions:
- The magnetic field [tex]\( B_1 \)[/tex] at a distance [tex]\( r_1 \)[/tex] from the wire is 0.1 T at 0.02 m.
We are asked to find the new magnetic field [tex]\( B_2 \)[/tex] at a new distance [tex]\( r_2 \)[/tex] of 0.01 m.
From the proportionality relationship, we can set up the following equation:
[tex]\[ B_1 \times r_1 = B_2 \times r_2 \][/tex]
Substitute the given values into this equation:
[tex]\[ 0.1 \, \text{T} \times 0.02 \, \text{m} = B_2 \times 0.01 \, \text{m} \][/tex]
Solve for [tex]\( B_2 \)[/tex]:
[tex]\[ B_2 = \frac{0.1 \, \text{T} \times 0.02 \, \text{m}}{0.01 \, \text{m}} \][/tex]
Calculate:
[tex]\[ B_2 = \frac{0.002 \, \text{T} \cdot \text{m}}{0.01 \, \text{m}} \][/tex]
[tex]\[ B_2 = 0.2 \, \text{T} \][/tex]
Therefore, the magnitude of the magnetic field 0.01 m from the same wire is:
[tex]\[ 0.2 \, \text{T} \][/tex]
Hence, the correct answer is [tex]\( 0.2 \, \text{T} \)[/tex].
