Pure copper wire of diameter = 2.0 mm has an electrical resistance, R, of 0.005 ohm/m (upside down U/m). p = 8900 kg/ m cubed, Cp= 440 J/(kg K). Passage of electrical current produces thermal energy generation = I triangle V= I squared R (J/sec).
Note that Units: Volts = J/coulomb; ohms = volts/amps; amps = coulombs/sec.
If the wire is perfectly insulated (no heat loss from the surface), calculate the dT/dt (degrees Celsius/sec) and the triangle T in one hour.
To calculate the rate of temperature change (dT/dt) and the change in temperature (ΔT) in one hour, we can use the equation for thermal energy generation, which is given by I²R (J/sec), where I represents the current in amps and R represents the resistance in ohms.
Given information:
Resistance (R) = 0.005 ohm/m
Diameter of wire = 2.0 mm
We are also given the formula for thermal energy generation:
I²R = J/sec
Since we want to calculate the rate of temperature change and the change in temperature, we need to relate the thermal energy generated to the change in temperature.
The formula for thermal energy generation can be written as:
I²R = mCpΔT/dt
Where:
m = mass of the wire
Cp = specific heat capacity of copper
ΔT/dt = rate of temperature change (dT/dt)
We can start by calculating the mass of the wire using its volume and density.
Since we know the diameter of the wire, we can calculate its radius (r) as:
r = diameter / 2 = 2.0 mm / 2 = 1.0 mm = 0.001 m
We can then calculate the cross-sectional area (A) of the wire using the formula for the area of a circle:
A = πr²
Now, let's calculate the cross-sectional area:
A = π(0.001 m)² = 0.00000314 m²
Next, we need to calculate the volume (V) of the wire per unit length:
V = A/m
Given that the wire is perfectly insulated, there is no heat loss from the surface, so the volume of the wire per unit length (V) remains constant throughout.
Now we can calculate the mass of the wire per unit length (m):
m = pV
Where:
p = density of copper = 8900 kg/m³ (given)
V = volume of wire per unit length
To find the volume (V) of the wire per unit length, we need to consider that the wire is cylindrical. So, V is simply the cross-sectional area (A) multiplied by the length of the wire.
Given information:
Length = 1 m (as the resistance is given per meter)
V = A * Length
Now we can find the mass of the wire per unit length (m):
m = p * V = p * A * Length
Next, we want to find the current (I) flowing through the wire. We can use Ohm's Law, which states that V = IR, where V represents voltage and I represents current.
In this case, we don't have the voltage (V) explicitly given. However, we can calculate V using the formula V = I * R, where I represents current and R represents resistance.
The resistance (R) is given as 0.005 ohms per meter. Since we are considering a 1-meter length of wire, the resistance (R) in this case is simply 0.005 ohms.
Now we can calculate the voltage (V) across the wire:
V = I * R
We can now rearrange this equation to solve for the current (I):
I = V / R
Since we are assuming the wire is perfectly insulated, there is no voltage drop along the length of the wire. Thus, the voltage (V) across the wire is constant.
Given information:
Voltage (V) = J/Coulomb (given)
Substituting the value of voltage (V) and resistance (R) into the equation, we can find the current (I):
I = V / R = Voltage / Resistance
With the current (I) and the resistance (R), we can calculate the thermal energy generation per unit time using the formula I²R.
Now we can calculate the rate of temperature change (dT/dt):
I²R = mCpΔT/dt
Rearranging the equation, we can isolate ΔT/dt:
ΔT/dt = I²R / (mCp)
Substituting the known values, we can calculate the rate of temperature change:
dT/dt = I²R / (mCp)
To calculate the change in temperature (ΔT) in one hour, we can multiply the rate of temperature change (dT/dt) by the time (1 hour = 3600 seconds):
ΔT = dT/dt * time
Substituting the known values, we can calculate the change in temperature:
ΔT = dT/dt * time
Please provide the necessary values for voltage (V) and time (t) to calculate the final temperature change.