The heating elements of an electric toaster is typically made of nichrome wire
(an alloy of nickel and chromium). As current passes through the wires, the wires heat up, thus toasting the toast. Estimate the overall resistance of a heating element which is 220 cm long and consists of nichrome wire with a diameter of 0.56 mm. The resistivity of nichrome is 110×〖10〗^(-8) Ω⋅m.(show work)
p = 110*10^-8 Ohm.m = 110*10^-6 ohm.cm.
L = 220 cm.
r = 0.028 cm.
A = pi*r^2 = 2.46*10^-3cm^2.
R = (110*10^-6*220)/(2.46*10^-3) = 9.84Ohms.
Why did the toast go to therapy? Because it had a heating element disorder!
To estimate the overall resistance of the heating element, we can use the formula for the resistance of a wire:
R = (ρ * L) / A
where R is the resistance, ρ is the resistivity of the material, L is the length of the wire, and A is the cross-sectional area of the wire.
First, we need to calculate the cross-sectional area (A) of the wire. Since the wire's diameter is given, we can find the radius (r) using the formula:
r = d / 2 = 0.56 mm / 2 = 0.28 mm = 0.28 × 10^(-3) m
Next, we can calculate the cross-sectional area:
A = π * r^2 = π * (0.28 × 10^(-3))^2 = 0.246 × 10^(-6) m^2
Now, we can plug the values into the resistance formula:
R = (ρ * L) / A = (110 × 10^(-8) Ω⋅m * 220 cm) / (0.246 × 10^(-6) m^2)
Converting the length from centimeters to meters:
R = (110 × 10^(-8) Ω⋅m * 2.2 m) / (0.246 × 10^(-6) m^2)
Simplifying:
R = (0.22 * 10^(-8) Ω⋅m) / (0.246 × 10^(-6) m^2)
Dividing the numerator and denominator by 10^(-8):
R = (0.22 Ω⋅m) / (2.46 × 10^(-2) m^2)
Simplifying further:
R = 0.00894 Ω ≈ 0.009 Ω
Therefore, the estimated overall resistance of the heating element is approximately 0.009 ohms.
To estimate the overall resistance of the heating element, we need to first calculate the resistance of a small section of the wire and then multiply it by the total number of sections.
The resistance of a wire can be determined using the formula:
R = ρ × (L/A)
Where:
R is the resistance
ρ (rho) is the resistivity of the material
L is the length of the wire
A is the cross-sectional area of the wire
In this case, we are given the length of the wire (220 cm) and the diameter of the wire (0.56 mm). However, the formula requires the cross-sectional area, not the diameter. We can calculate the cross-sectional area (A) using the formula:
A = π × (r^2)
Where:
A is the cross-sectional area
π (pi) is a mathematical constant, approximately equal to 3.14159
r is the radius of the wire
To calculate the radius from the diameter, we divide the diameter by 2:
r = d/2 = 0.56 mm / 2 = 0.28 mm = 0.28 × 10^(-3) m
Now we can substitute the values into the formulas to calculate the resistance:
A = π × (0.28 × 10^(-3))^2 = 0.2464 × 10^(-6) m^2
R = (110 × 10^(-8)) × (220 / 0.2464 × 10^(-6))
R = 110 × 10^(-8) × 8.927 m
R ≈ 98 × 10^(-6) Ω
The overall resistance of the heating element is approximately 98 × 10^(-6) Ω.
R = pL/A.
p = 110*10^-8 Ohm.m = 110*10^-6 ohm.cm.
L = 220 cm.
r = 0.056 cm.
A = pi*r^2 = 3.14*0.056^2 = 9.85*10^-3 cm^2.
R = (110*10^-6*220)/(9.85*10^-3) = 2.46 Ohms.