To find the current density and the diameter of the nichrome wire, we can start with the definitions and the formulae involved.
a) Current Density: Current density (J) is defined as the current (I) flowing through a wire per unit cross-sectional area (A) of the wire. It can be calculated using the formula:
J = I / A
First, we need to find the cross-sectional area of the wire. The formula for the area of a circle (cross-section of the wire) is:
A = π * (d/2)²
But first, we will find the resistivity of nichrome, which is approximately 1.10 × 10-6 Ω·m and use it in conjunction with the voltage and current.
Using Ohm’s Law:
V = I * R
Where R (resistance) can be calculated using:
R = ρ * (L / A)
As we know:
V = I * (ρ * (L / A))
Let’s rearrange this to express A:
A = ρ * (L / V) * I
Plugging in the values: ρ = 1.10 × 10-6 Ω·m, L = 0.19 m, and V = 1.5 V:
A = (1.10 × 10-6) * (0.19) / (1.5 / 2.6)
Upon simplification, we can find A, and then substitute A back into the current density formula to find J:
J = I / A
b) Wire Diameter: Once we have the cross-sectional area A, we can solve for the diameter (d). From the area formula:
A = π * (d/2)²
Rearranging for diameter gives:
d = 2 * sqrt(A / π)
Substituting the calculated area back into this equation will provide us the diameter of the wire.
To summarize: the current density can be calculated based on the current and area, and the diameter can be calculated from the area determined from resistivity and resistance. These equations give a clear path toward the solutions. Make sure to compute the respective values accurately for final results.