A rectangular piece of gold is 2.5 cm long and 2.6 cm wide. How deep is the piece when it displays a resistance of R = 1.39 μω for a current that flows along the depth-direction?
To determine the depth of the rectangular piece of gold, we can use the formula for resistance:
R = ρ * (L / A)
Where:
R is the resistance,
ρ (rho) is the resistivity of the material,
L is the length along the direction of the current flow,
A is the cross-sectional area perpendicular to the direction of the current flow.
We are given:
Length of the rectangular piece (L) = 2.5 cm
Width of the rectangular piece (W) = 2.6 cm
Resistance (R) = 1.39 μω (micro-ohms)
First, we need to find the cross-sectional area (A) of the rectangular piece. Since we are considering the current flowing along the depth-direction, the cross-sectional area will be the product of the length (L) and the width (W):
A = L * W
A = 2.5 cm * 2.6 cm
A = 6.5 cm²
Next, we need the resistivity of gold (ρ). The resistivity of gold is typically around 2.44 x 10^(-8) Ωm (ohm-meters).
Now we can substitute these values into the resistance formula:
1.39 μω = (2.44 x 10^(-8) Ωm) * (2.5 cm / 6.5 cm²)
To simplify the units, we need to convert the length from centimeters to meters:
2.5 cm = 0.025 m
Now we can substitute the values again and solve for ω:
1.39 μω = (2.44 x 10^(-8) Ωm) * (0.025 m / 6.5 cm²)
Simplifying the equation, we get:
1.39 * 10^(-6) Ω = 3.77 * 10^(-8) Ωm / cm²
To solve for ω, we divide both sides of the equation by 3.77 * 10^(-8) Ωm / cm²:
(1.39 * 10^(-6) Ω) / (3.77 * 10^(-8) Ωm / cm²) = ω
The resulting value of ω will give us the depth of the rectangular piece of gold when it displays a resistance of R = 1.39 μω for a current that flows along the depth-direction.