A cylindrical copper cable 1.50Km long is connected across a 220.0V potential difference.
(a) What should be its diameter so that it produces heat at a rate of 50.0W?
(b) What is the electric field inside the cable under these conditions?
First, the value of the length of the cable will be converted to meters.
Therefore, the length of the cable is L = 1.5*10^3 m = 1500 m.
We know, from enunciation, the value of potential difference:V = 220 volts
The power is given and it's value is of P = 50 Watts
P = V^2/R => R = V^2/P
Since the area of the section of the cable is circular, w'ell recall the formula for the area of the circle:
A = pi*r^2 (1)
A = p*L/R (2), where p = 1.72/10^8 ohm/m
We'll equate (1) and (2) and we'll get:
pi*r^2 = p*L*P/V^2
r^2 = p*L*P/V^2*pi
r = sqrt(p*L*P/V^2*pi)
Diameter is d = 2*r = 2*sqrt(p*L*P/V^2*pi)
d = 2*sqrt1.72*10^-10*15*5*10^3/484
d = 2*sqrt 1.72*10^-7*75/22
d = 1.0325/100
d = 0.010325 meters
The diameter of the cable is of d = 0.010325 meters, such as it is producing heat at a rate of 50W.
(a) Well, to calculate the diameter of the copper cable, we need to make sure it's hot enough to impress your neighbors. Let's use some electricity humor to solve it!
To find the diameter, we need to consider the heat generated. So, if the cable is producing heat at a rate of 50.0W, we need to make sure it's not too hot to handle. Copper has a resistivity of about 1.68 x 10^-8 Ω.m.
Using the formula for power:
P = (pi * (d^2) * L * σ * |V|^2) / (4 * R^2)
where P is the power, pi is pi (obviously), d is the diameter, L is the length, σ is the resistivity, |V| is the potential difference, and R is the radius (half the diameter).
We can rearrange the formula to solve for d:
d^2 = (4 * P * R^2) / (pi * L * σ * |V|^2)
d^2 = (4 * 50.0 * (R^2)) / (pi * 1.50 * 10^3 * 1.68 * 10^-8 * (220.0^2))
Now, let's grab our trusty calculators and compute that:
d^2 ≈ 6.608 x 10^-9
Taking the square root of both sides, we find:
d ≈ 8.12 x 10^-5 meters
So, the diameter of the copper cable should be approximately 8.12 x 10^-5 meters to produce heat at a rate of 50.0W. Just be careful not to touch it if you prefer having unburnt fingers.
(b) Ah, electric fields inside cables! It's like a circus inside there! To find the electric field, we can use the formula:
E = |V| / d
Substituting the values:
E = 220.0 / (8.12 x 10^-5)
Now let's do some math:
E ≈ 2.71 x 10^6 V/m
So, the electric field inside the cable under these conditions is approximately 2.71 x 10^6 volts per meter. It's shocking, in a good way! Just remember to wear your rubber gloves when dealing with those electric fields. Safety first, my friend!
To find the diameter of the copper cable and the electric field inside, we can use the following formulas:
(a) The formula to calculate the power (P) dissipated as heat in a resistor is given by:
P = (pi * (d^2) * L * rho * E^2) / (4 * R)
Where:
P = Power dissipated (50.0 W)
pi = Pi (approximately 3.14)
d = Diameter of the cable (to be determined)
L = Length of the cable (1.50 km or 1500 m)
rho = Resistivity of copper (approximately 1.7 x 10^-8 ohm*m)
E = Electric field strength (to be determined)
R = Resistance of the cable
Since we don't know the resistance of the cable, we will need to find it first.
The resistance of a wire can be calculated using the formula:
R = (rho * L) / A
Where:
R = Resistance
rho = Resistivity of copper (1.7 x 10^-8 ohm*m)
L = Length of the cable (1500 m)
A = Cross-sectional area of the cable (to be determined)
From the formula for the area of a cylinder:
A = pi * (d^2) / 4
Substituting this value of A into the resistance equation, we can solve for R:
R = (rho * L) / (pi * (d^2) / 4)
Now, we can substitute the value of R into the power equation and solve for d.
50.0W = (pi * (d^2) * 1500m * 1.7 x 10^-8 ohm*m * (E^2)) / (4 * R)
Now, we can substitute the given values and solve for d.
(b) The electric field inside the cable can be calculated using Ohm's Law:
E = V / d
Where:
E = Electric field strength (to be determined)
V = Potential difference (220.0 V)
d = Diameter of the cable (to be determined)
Let's solve these equations step-by-step:
(a) Calculation of the Diameter:
Step 1: Calculate the Resistance (R):
R = (1.7 x 10^-8 ohm*m * 1500m) / (pi * (d^2) / 4)
Step 2: Substitute R into the Power equation:
50.0W = (pi * (d^2) * 1500m * 1.7 x 10^-8 ohm*m * (E^2)) / (4 * (1.7 x 10^-8 ohm*m * 1500m) / (pi * (d^2) / 4))
Simplify the equation to solve for d.
(b) Calculation of the Electric Field:
E = 220.0V / d
Now, calculate the value of d using the equation found in part (a), then substitute it into the equation for the electric field.
Please note: Due to the complexity of the calculations, it is recommended to use a calculator or spreadsheet software for precise calculations.
To solve part (a) of the problem, we need to use the formula for the power dissipated in a resistor, which is given by:
Power = (Current)^2 * Resistance
In this case, the copper cable acts as a resistor, and the power dissipated is given as 50.0W. We also know that the potential difference across the cable is 220.0V.
To find the current passing through the cable, we can use Ohm's Law, which states that the current (I) is equal to the potential difference (V) divided by the resistance (R):
Current = Voltage / Resistance
The resistance of a cylindrical cable can be calculated using the formula:
Resistance = resistivity * (Length / Area)
Where resistivity is a constant property of copper, Length is the length of the cable, and Area is the cross-sectional area of the cable.
To calculate the diameter of the cable, we need to find the cross-sectional area. The formula for the area of a circle is:
Area = π * (Radius)^2
To solve for the radius, we need to calculate the diameter first.
Now, let's break down the steps to solve part (a) of the problem:
Step 1: Calculate the resistance of the copper cable.
- Use the resistivity of copper (given in reference tables) to find the resistance of 1 meter of copper cable using the formula: Resistance = resistivity * (Length / Area).
Step 2: Calculate the current passing through the cable.
- Use Ohm's Law: Current = Voltage / Resistance.
Step 3: Calculate the diameter of the cable.
- Rearrange the formula for the area of a circle to solve for the radius: Radius = sqrt(Area / π).
- Double the radius to get the diameter.
Now, to solve part (b) of the problem, we need to calculate the electric field inside the cable. The electric field (E) inside a cylindrical cable is given by:
Electric field = (Voltage) / (Length * ln(diameter / radius))
- Use the given values for the potential difference and the length of the cable to calculate the electric field.