Question Video: Using Ohm’s Law to Calculate the Number of Electrons That Pass through a Heater Physics • Third Year of Secondary School

Video Transcript

An electric heater that has a resistance of 17.7 ohms is connected to a 230-volt power supply. Given that the electron charge 𝑒 equals 1.6 times 10 to the negative 19 coulombs, how many electrons will pass through the heater every minute? Give your answer in scientific notation to two decimal places. (A) 4.87 times 10 to the 21 electrons, (B) 1.53 times 10 to the 24 electrons, (C) 2.89 times 10 to the 19 electrons, (D) 1.35 times 10 to the 18 electrons.

In order to solve this problem, we will first need to use Ohm’s law to find the current through the heater. Ohm’s law can be written as 𝑉 equals 𝐼 times 𝑅, where 𝑉 is the potential difference, 𝐼 is the current, and 𝑅 is the resistance. We are concerned with finding the current, so let’s rearrange the equation to make current the subject. We can do this by dividing both sides of the equation by the resistance 𝑅. The resistances on the right-hand side cancel each other out, and we are left with the current 𝐼 equals the potential difference 𝑉 divided by the resistance 𝑅. By substituting the values of potential difference and resistance, we can work out that the current is equal to 230 volts divided by 17.7 ohms. This equals 12.99435 et cetera amperes.

Next, we need to find the number of electrons that pass through the heater in a minute. Now, let’s recall that current is the flow of electric charge. Current is defined using the equation 𝐼 equals 𝑄 over 𝑡, where 𝐼 is the current, 𝑄 is the total charge moving past a point, measured in coulombs, and 𝑡 is the time taken for that amount of charge to move, and this time is measured in seconds. We can find the total charge 𝑄 by multiplying both sides of this equation by the time 𝑡. We have that 𝑄 equals 𝐼 multiplied by 𝑡. Remember that the time 𝑡 has to be in units of seconds. To convert our time of one minute into seconds, we need to multiply it by 60. One minute is equal to 60 seconds.

Now, we simply substitute the current, 12.99435 amperes, and the time, 60 seconds, into our expression for the total charge 𝑄. We find that 779.66101 et cetera coulombs of charge pass through the heater every minute. Since the total charge is provided by electrons, the value of 𝑄 must be equal to the number of electrons multiplied by the charge carried by each electron. We can write this as the expression 𝑄 equals 𝑁 times 𝑒, where 𝑁 is the number of electrons and 𝑒 is the electron charge, 1.6 times 10 to the negative 19 coulombs.

If we divide both sides of this equation by the electron charge, we can determine the number of electrons that pass through the heater every minute. So when we substitute the values of total charge and the electron charge, we get the number of electrons 𝑁 is equal to 779.66101 et cetera coulombs divided by 1.6 times 10 to the negative 19 coulombs, which equals 4.87288 et cetera multiplied by 10 to the 21 electrons.

The final bit that we need to complete this question is to give the answer to two decimal places. We will take the number before the decimal place and the three numbers after the decimal place to get 4.872 multiplied by 10 to the 21 electrons. Then, we look at the third number after the decimal place to see if we need to round down or round up. We see that this third number is two, so we should round down, which leaves the answer to be 4.87 times 10 to the 21 electrons. We can see that this corresponds to option (A).

Therefore, option (A) is the correct answer. The number of electrons that will pass through the heater every minute is 4.87 times 10 to the 21 electrons.