a toaster with 4m length of nichrome(resistivity is 150×10^-8 and temperature coefficient of resistivity is 0.4×10^-3/¤c) wire as its heating element is rated at 1050w and operates on 220v.the operating temperature of this is 320deg celcius. what is the radius of the wire?

To find the radius of the wire, we can use the formula for the resistance of a wire:

R = (ρ x L) / A

Where:
R = Resistance
ρ = Resistivity
L = Length of the wire
A = Cross-sectional area of the wire

First, let's calculate the resistance of the wire using the power and voltage:

P = V^2 / R

Given that the power (P) is 1050 W and the voltage (V) is 220 V, we can rearrange the formula to solve for resistance (R):

R = V^2 / P
R = (220)^2 / 1050
R = 48400 / 1050
R ≈ 46.095 ohms

Next, we can calculate the change in resistance due to the change in temperature:

ΔR = R x α x ΔT

Where:
ΔR = Change in resistance
α = Temperature coefficient of resistivity
ΔT = Change in temperature

Given that the temperature coefficient of resistivity (α) is 0.4 × 10^-3 / °C and the change in temperature (ΔT) is (320 - 20) = 300 °C, we can calculate the change in resistance:

ΔR = 46.095 x 0.4 × 10^-3 / °C x 300 °C
ΔR ≈ 5.5274 ohms

Now, let's calculate the resistance at the operating temperature:

R_operating = R + ΔR
R_operating = 46.095 + 5.5274
R_operating ≈ 51.6224 ohms

Finally, we can calculate the radius of the wire using the resistance and resistivity:

R_operating = (ρ x L) / A

Rearranging the formula to solve for the area (A), we have:

A = (ρ x L) / R_operating

Given that the resistivity (ρ) is 150 × 10^-8 Ω·m and the length (L) is 4 m, we can calculate the radius (r):

A = π x r^2
r = √(A / π)

Substituting the values, we have:

r = √((150 × 10^-8 x 4) / (51.6224 x π))
r = √(6 × 10^-7 / (51.6224 x π))
r ≈ √(1.159 x 10^-9) ≈ 3.4082 x 10^-5 m

Therefore, the radius of the wire is approximately 3.4082 x 10^-5 meters.

To find the radius of the wire, we need to use the formula for the resistance of a wire:

R = (ρ * L) / A

Where:
R is the resistance of the wire,
ρ is the resistivity of the wire material,
L is the length of the wire, and
A is the cross-sectional area of the wire.

First, we can calculate the resistance of the wire using the power and voltage values:

R = P / (V^2)

Where:
P is the power rating of the toaster, and
V is the voltage at which it operates.

Given that the power rating is 1050W and the operating voltage is 220V, we can plug these values into the formula:

R = 1050 / (220^2)

R = 1050 / 48400

R = 0.0216942 ohms

Next, we need to calculate the resistivity at the operating temperature. The change in resistivity with temperature can be calculated using the formula:

∆ρ = ρ * α * ∆T

Where:
∆ρ is the change in resistivity,
ρ is the resistivity at room temperature,
α is the temperature coefficient of resistivity, and
∆T is the change in temperature.

Given that the room temperature is 25°C and the operating temperature is 320°C, we can calculate the change in resistivity:

∆T = (320 - 25)

∆T = 295°C

∆ρ = ρ * α * ∆T

∆ρ = (150 × 10^-8) * (0.4 × 10^-3/°C) * 295

∆ρ = 150 × 0.4 × 10^-8 × 0.4 × 10^-3 × 295

∆ρ = 0.01776 ohms.m

Now, we can find the resistivity at the operating temperature:

ρ_operating = ρ + ∆ρ

ρ_operating = 150 × 10^-8 + 0.01776

ρ_operating = 1.576 × 10^-5 ohms.m

Lastly, we can find the cross-sectional area of the wire using the formula:

A = (π * r^2)

Where:
A is the cross-sectional area,
r is the radius of the wire,
and π is a constant approximately equal to 3.14.

To find the radius, we rearrange the formula:

r = √(A / π)

To calculate A, we can use the formula:

A = (ρ_operating * L) / R

Substituting the known values:

A = (1.576 × 10^-5 * 4) / 0.0216942

A = 2.892 × 10^-3 m^2

Now, we can calculate the radius:

r = √(2.892 × 10^-3 / 3.14)

r = √(0.000920382166)

r = 0.0303 m

Therefore, the radius of the wire is approximately 0.0303 meters.