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# Solved example: Finding current & voltage in a circuit

## Video transcript

we have three resistors connected as shown across a 50 volt supply the question which have not written down to save space is to find the voltage across each resistor and to find the current through each resistor now before we start solving this let's quickly go through a common mistake that I would do while solving problems like this what I would do is apply Ohm's law to each resistor directly so remember Ohm's law Ohm's law says V equals I times R and what I am thinking over here what I used to think over here is I already know the voltage is 50 so then for 2 ohm resistor to calculate the current here I would substitute RS to V is 50 calculate the current then for 40 ohm resistor I would put V is 50 that's already given R is 40 calculate the current same thing over here and we are done we now know current through each resistor but you understand that's wrong why is that wrong well that's because when we apply warms law V which is the voltage is the potential difference across that resistor for example if I'm using this for 2 ohm resistor then I need to know what's the potential difference across 2 ohms but I don't know what's the potential difference across 2 ohms 50 world is the potential difference across these two points this point has the same voltage as this point and this point has the same voltage at this point which means I know the potential difference across this and this point I don't know the potential difference across 2 ohms so I can't apply it for 2 ohms I don't know the potential difference across 10 ohms I can't apply it for 10 or I don't know hear it even here and that's why I can't directly solve the problem if you substitute V as 50 for each resistor we are implying that 50 volt is the potential difference across each resistor which is clearly wrong and that's why we can't do it that way all right so what's the what's the correct way to do this the correct way to do this is since I know the voltage across these two points I to first create what is the equivalent resistance of these three I need to replace these three resistance resistors with one single resistor once I have done that then I can go ahead and apply Ohm's law and calculate it so I need to reduce this circuit and we have seen how to reduce circuits like this in a previous video so it would be a great idea to first pause and see if you can try this yourself all right let's do this so all we need to do is identify resistors in series and in parallel let's start with two and ten do you think they are in series they look like they're in series but are they in series the answer is no the way I like to test where the two resistors are in series or not is remember that they did it to have the same current flowing through them right that's the definition of series so I would imagine a small current flowing over here and see if that entire current flows here if it does they are in series if not they're not in series so imagine some current flowing here now as that current goes forward notice there's a branch because of that some current might flow up and the rest of the current will flow here and as a result the current here and here need not be the same so they are not in series with each other but if you look at these two resistors they are in parallel how do I check whether two resistors are in parallel they need to have the same voltage across them let's quickly check that this point has the same voltage at this point because there are no resistors in between this point has the same voltage as this point because there are no resistors in between a wire will always have the same voltage anywhere we're assuming the wires don't have any resistances and when there is no resistance the potential difference is always zero within a wire across any two points in a wire so the voltage is the same and so what is this voltage the potential difference here is the same as potential difference here and therefore they are in parallel with each other but for example if there was a resistor over here then these two voltage these two points won't have the same voltage and then they wouldn't be in parallel but anyways these are in parallel and so we can go ahead and replace these resistor with an equivalent resistance so how do we calculate equivalent resistance in parallel well the formula for equivalent resistance in parallel is 1 over R equivalent is going to be 1 over R 1 which is going to be 1 or 40 for us 1 over R 1 plus 1 over R 2 which is going to be 1 over 10 1 over 10 so let's solve this we have a common denominator of 40 this will be 1 plus I have to multiply this by 4 to get 40 so multiply the numerator also by 4 that gives me 5 or 40 and remember this is 1 over R equivalent so our equivalent would be let's write that down the reciprocal of this 40 over 5 and that is 8 ohms so what we have calculated is that these two resistors connected in parallel can be replaced by a single resistor of 8 ohm and nothing will change it's the current and everything in the circuit will remain the same the current in the circuit and the voltage everything will remain the same so let's go ahead and do that so what we'll do is I'll keep the rest of the circuit as it is so let's draw the rest of the circuit as it is but replace this combination with a single resistor of 8 ohms there it is and now these two resistors are in series with each other how do we check whether they are in series or not how do we confirm they need to have the same current flowing through them so let's imagine a current going here and notice all the current will flow here there are no branches right now and therefore they are in series and when resistors are in series the equivalent resistance is just the sum of the individual resistances and therefore I will not write it down the equivalent over here would be just the sum of these two a plus 2 be 10 ohms and so again we can now replace these two resistors with a single resistor of 10 ohms and keep the rest of the circuit as it is so let's do that here it is and we are done with reduction because we have reduced this circuit to a single resistor and now I know the voltage across these two points which is the same as the voltage across this point now I know this voltage is 50 volts so I know let's write it down this point the voltage between these two points is 50 volts I know that this is 50 volts and so for this equivalent resistance I can now go ahead and apply Ohm's law and calculate the current through this resistance and that's what we'll do next so let's get rid of this to make some space and let's apply Ohm's law here so we know V is 50 V is 50 that's equal to I times R R is 10 R is 10 so I is 50 divided by 10 that's going to be 5 amperes so I is 5 so the current in this circuit is going to be 5 this is positive this is negative so the current flows from positive terminal to the negative terminal and so that's 5 amperes so the current flowing through this resistor is 5 amperes but hold on our original question is to calculate the current through each of these three resistors and the voltage across these three resistors but what we have done now is calculate the current in this equivalent resistance how do we get from here to there well now the trick is we go backwards from here that's why it's important to write down each step so here's what I mean if we go from if we go back from here to here this 10 ohm splits as 2 and 8 and this splitting is a series splitting that's how I like to think about it and remember in series the current is the same so whatever is flowing here the same current must flow through this resistor and this resistor as well because in series current remains the same so the moment I know that the current here is 5 amps I also know that the current here and the current here of course it must be the same current that is also 5 amperes and once I know the current the next thing I will do immediately is to calculate the voltage across those resistors but again applying Ohm's law so over here notice I know the current is 5 the resistance is 2 V equals IR so the voltage here must be 10 volts let's use the same color so this world voltage across this resistance must be 10 volts and similarly the voltage across this resistance IR 5 times 8 must be 40 volts and just to confirm notice 10 and 40 adds up to give us a total of 50 makes sense because from here to here the total voltage must be 50 volts what's the next step let's go backwards we are already done with this 2 ohms we already know if this is 5 amps and we know the voltage here is 10 volt that part is already done what to do here well now this 8 ohm splits as 40 and 10 as a parallel combination it's a parallel split as I would like to think about it and remember in parallel they have the same voltage so whatever is the voltage here must be the same voltage over here so immediately I know the voltage across this must be 40 volts and the voltage here must also be 40 volts oops wrong color let's use the same color so the voltage here must also be 40 words it's a little shabby but hopefully the color helps you identify our differentiate between them and now that I know the world age again apply Ohm's law this time to calculate the current so that's the whole game over here if you know voltage you calculate the current if you know the current you calculate the voltage so in this resistor the resistance is 10 voltage is 40 so I is V over R so 40 divided by 10 that's going to be 4 amps so current here is going to be 4 amps and over here 40 divided by 40 is going to be 1 amp 1 amp and again just to check still notice that the 5 amp is getting split as 1 amp and for M 4 plus 1 is 5 so again this confirms that whatever we did is right and we have now solved our problem because we now know the current through each resistor and we also know the voltage across each resistor and so to summarize whenever we have a question like this where we have a bunch of resistors connected in connected in some combination across some voltage source and they're asked to calculate the the current and the voltage across each one first we'll reduce it to a single resistance then we'll calculate the current in that through that resistance and the voltage across that resistance and then we'll keep back will keep backtracking when we go back if the resistors split as series then we know the current must be the same and then we know the current next step would be to calculate the voltage if we go back and we find they split as parallel resistors then the world age is the same then we use Ohm's law to calculate the current and that's how you keep on backtracking regardless of how complicated the circuit is as long as you can reduce it to a single resistor and you write down all the steps in between that's important otherwise it becomes a little bit difficult to do this as long as you have written all the steps as in you've drawn all the sub circuits in between we can always go back and keep doing this calculate the voltage and calculate the current you