Final answer:
To find the radius of the wire, we can use the formula for energy dissipated as heat in a circuit. By using the known values of the energy loss, current, and length of the wire, we can calculate the resistivity and cross-sectional area of the wire, and finally determine the radius.
Step-by-step explanation:
To find the radius of the wire, we can use the formula for energy dissipated as heat in a circuit:
E = I^2 * R * t
Where E is the energy dissipated (in Joules), I is the current (in Amperes), R is the resistance (in Ohms), and t is the time (in seconds).
In this case, we are given the energy loss (E = 3.25 MJ = 3.25 × 10^6 J) and the current (I = 2.50 A) for the first second (t = 1 s).
Rearranging the formula, we can solve for the resistance:
R = E / (I^2 * t)
Substituting the given values:
R = (3.25 × 10^6 J) / (2.50 A)^2 * (1 s)
Simplifying gives:
R = 52000 Ω
Since the resistance of a wire is given by:
R = ρ * (L / A)
Where ρ is the resistivity of the material (for copper it is approximately 1.7 × 10^-8 Ωm), L is the length of the wire, and A is the cross-sectional area of the wire.
Using the given length of the wire (L = 1.25 m), we can rearrange the formula to solve for the cross-sectional area:
A = ρ * (L / R)
Substituting the known values:
A = (1.7 × 10^-8 Ωm) * (1.25 m / 52000 Ω)
Simplifying gives:
A = 4.09 × 10^-11 m^2
Since the wire is cylindrical, the cross-sectional area is given by:
A = π * r^2
Where r is the radius of the wire. Rearranging the formula and solving for r:
r = sqrt(A / π)
Substituting the calculated value for A:
r = sqrt(4.09 × 10^-11 m^2 / π)
Calculating gives:
r = 1.02 × 10^-6 m = 1.02 mm
Therefore, the radius of the wire is 1.02 mm.