A rectangular piece of gold is 3.8 cm long and 2.5 cm wide. How deep is the piece when it displays a resistance of R = 1.35 μω for a current that flows along the depth-direction?
Use the formula that relates resistance to resisitivity. You will have to look up the resisitivity (or conductivity) of gold
To find the depth of the gold piece, we need to use its resistance and the dimensions of the piece. The resistance of an object is given by the formula:
R = (ρ × L) / A
Where:
R = resistance
ρ = resistivity of the material
L = length of the object
A = cross-sectional area of the object
In this case, we know the resistance (R = 1.35 μΩ), the length (L = 3.8 cm), and the width (W = 2.5 cm). We can rearrange the formula to solve for the resistivity (ρ), and once we have that value, we can calculate the depth.
1. Solve for the resistivity (ρ):
R = (ρ × L) / A
Rearrange the formula to solve for ρ:
ρ = (R × A) / L
2. Calculate the cross-sectional area (A):
Since the object is rectangular, the cross-sectional area is the length (L) multiplied by the width (W):
A = L × W
3. Substitute the given values into the formula:
A = 3.8 cm × 2.5 cm
= 9.5 cm²
4. Convert the resistance from micro-ohms to ohms (μω to Ω):
1.35 μΩ = 1.35 x 10^-6 Ω
5. Substitute the values of R, A, and L into the resistivity formula:
ρ = (1.35 x 10^-6 Ω) × (9.5 cm²) / (3.8 cm)
6. Calculate the resistivity (ρ).
7. Once you have the resistivity, you can determine the depth using the resistivity formula:
ρ = R × (A / L)
Substitute the known values:
1.35 x 10^-6 Ω = ρ × (9.5 cm² / L)
Solve for L, the depth of the gold piece.