A generating station is producing 1.2 x 106 W of power that is to be sent to a small town located 4.2 km away. Each of the two wires that comprise the transmission line has a resistance per length of 5.0 x 10-2/km. (a) Find the power lost in heating the wires if the power is transmitted at 1200 V. (b) A 100:1 step-up transformer is used to raise the voltage before the power is transmitted. How much power is now lost in heating the wires?
The total resistance of the wires is Rwire = 4.2*0.05 = 0.21 ohms.
The current flowing in the wires when delivering 1.2*10^6 W is P/V
= 1.2*10^6 W/1200 = 1000 A
Poower lost in wires
= I^2*Rwire = 210,000 W
With a 100:1 stepup transformer, I is 100 x less. The power is delivered the same but the loss in the wires is 10^4 times less, or 21 W
To find the power lost in heating the wires, we can use the formula for power loss in a transmission line, which is given by:
P_loss = I^2 * R * L,
where P_loss is the power loss in watts, I is the current in amperes, R is the resistance per length in ohms/km, and L is the length of the transmission line in km.
(a) Find the power lost in heating the wires if the power is transmitted at 1200 V:
- First, we need to calculate the current flowing through the transmission line. We can use Ohm's law, which states that:
V = I * R * L,
where V is the voltage in volts, I is the current in amperes, R is the resistance per length in ohms/km, and L is the length of the transmission line in km.
- Rearranging the equation, we have:
I = V / (R * L),
where I is the current in amperes.
- Now, we can substitute the given values into the equation:
V = 1200 V,
R = 5.0 x 10^-2 ohms/km,
L = 4.2 km.
- Plugging in the values, we get:
I = 1200 V / (5.0 x 10^-2 ohms/km * 4.2 km) = 5714.29 A.
- Next, we can calculate the power loss using the formula:
P_loss = I^2 * R * L.
- Plugging in the values, we get:
P_loss = (5714.29 A)^2 * (5.0 x 10^-2 ohms/km) * 4.2 km = 1.225 x 10^6 W.
Therefore, the power lost in heating the wires is 1.225 x 10^6 W.
(b) A 100:1 step-up transformer is used to raise the voltage before the power is transmitted. How much power is now lost in heating the wires?
- When the voltage is stepped up by a factor of 100, the current will be reduced by the same factor. So, the new current (I_new) can be calculated as:
I_new = I / 100 = 5714.29 A / 100 = 57.14 A.
- The power loss with the stepped-up voltage can now be calculated using the formula:
P_loss_new = I_new^2 * R * L.
- Plugging in the values, we get:
P_loss_new = (57.14 A)^2 * (5.0 x 10^-2 ohms/km) * 4.2 km = 29,478 W.
Therefore, the power lost in heating the wires with the use of the step-up transformer is 29,478 W.
To find the power lost in heating the wires, we can use the formula for power in a circuit, which is given by P = I^2 R, where P is the power, I is the current, and R is the resistance.
(a) First, let's calculate the current flowing through the transmission line.
To do that, we need to find the total resistance of the transmission line. As there are two wires, each with a resistance per length, we need to multiply that value by the total length of the transmission line.
Total resistance = (resistance per length) x (total length)
= (5.0 x 10^-2 / km) x (4.2 km)
Now, we can calculate the total current using Ohm's Law, V = IR, where V is the voltage and R is the resistance.
Substituting the values, we get V = IR
1200 V = I x (total resistance)
Solving for I:
I = 1200 V / (total resistance)
Now, we can calculate the power lost in the wires using the formula P = I^2 R. Since we have the current and the resistance, we can substitute them in and calculate the power.
P = (I^2) x (total resistance)
(b) For the second part, where a step-up transformer is used to raise the voltage, the transmission voltage is increased by a factor of 100.
So, the new transmission voltage is 1200 V x 100 = 120,000 V.
Using the new transmission voltage, we can repeat the calculations from part (a) to find the new current and the new power lost in the wires.
Remember to use the updated voltage value in the calculations for part (b).
Note: It's important to note that this calculation assumes that all the power generated is transmitted to the small town. In reality, there may be other factors that affect power loss during transmission, such as line losses and inefficiencies.