Current sensitivity (with numerical) of a galvanometer

let's introduce the idea of current sensitivity of a galvanometer and see why we should care about that in a previous video we built a moving coil galvanometer and the whole idea was when you pass some current through the coil this current coil behaves like a tiny magnet and tiny magnets inside external magnetic fields experience a torque making this coil twist and this is the expression that we found for this torque and the important thing over here was this angle between the magnetic moment and the external magnetic field will always be 90 degrees at any position it's 90 degrees over here 90 degrees over here 90 degrees over here will always stay 90 degrees because the field is radial and at the same time we have coil springs which don't like to get twisted so as they get twisted they try to untwist producing a counter torque in the opposite direction and this counter torque is found to be proportional to the angle through which this coil twists and at equilibrium the two torques are exactly equal to each other and as a result we see that the twist produced is proportional to the current giving us a linear galvanometer exactly what we want and if you need a refresher on this feel free to go back our previous video on moving coil galvanometers we have talked about this in great detail but now let's rearrange this equation to give us phi the deflection divided by the current i if i rearrange this we will end up with on the right hand side we have this n a times b divided by the c the spring constant and now this term let me keep it to the right this term is what we call the current sensitivity of a galvanometer let's see why let's see what it means first of all look at the units of this because there's a deflection on the numerator and current in the denominator the unit becomes amperes sorry degrees per ampere so let's take some values imagine in our galvanometer this value was a thousand degrees per ampere then what does that mean well it would mean that if you were to put one amp of current through this coil it will tend to produce a deflection of a thousand degrees and can you imagine 1000 degrees like it tends to produce three turns which is just not possible it gets text stuck somewhere which means this galvanometer cannot handle one amp here something will break over here which means we say that this is a very sensitive galvanometer so you have high sensitivity high sensitivity which means you use this calvino meter to measure very small current so maybe use this to measure milliamps of currents or hundreds tens of milliamps of current so measure small currents measure small currents on the other hand what if this value was low maybe i don't know maybe say 0.1 degree per ampere what does this mean well this would mean now if you put one amp of current you get a very tiny amount of deflection so it's not very sensitive to current so you can use this to measure a lot of current let's say about hundreds of amps of current so this is low sensitivity so this is low sensitivity and use this to measure high currents measure high currents so clearly this number tells us how sensitive the output of the galvanometer the deflection of the galvanometer is to the input current so high sensitivity is kind of like when i was a child when somebody would say a little bit of mean things to me i would just cry out loud very high output very very sensitive and low sensitivity is like how i'm right now i'm a little bit mature and so even if you tell a lot of mean things do mean to me i won't produce a lot of output i'm very less sensitive to mean things now and so different applications would require different sensitivity values and that brings us to the question how do we control the sensitivity of our galvanometer now and for that we look on the right hand side so if you increase these the sensitivity tends to increase on the other hand if you increase this the sensitivity would tend to decrease let's see why well if you increase the number of turns or the area of this coil or the magnetic field strength by producing say stronger magnets in all these cases the torque that is produced tends to increase think about it the torque given due to a current tends to increase you to these values and if the torque tends to increase you tend to get more deflection more output and as a result you tend to get more sensitivity does that make sense on the other hand what happens when you increase the value of c what is this c again remember c is the spring constant think of it as it tells you how stiff the spring is so if you increase the value of c you are making the spring stiffer if you make the spring stiffer it tends to generate a large counter torque which means it will not allow a lot of output it will not allow a lot of deflection and that means it tends to decrease the sensitivity so by controlling these values you can get whatever sensitivity you want now let's see if we can apply this in a numerical so let's keep this formula at the side and here is a numerical why don't you pause the video read this thing and see if you can try and answer this question yourself first all right what i'll do is i'll color code things so we can understand what is what so here's my color coded version we are given a galvanometer has a coil of area so its area is given the number of turns is given to us its magnetic field is given we are also given what the spring constant is and we are given what the maximum deflection a galvanometer can accommodate so the that's the maximum deflection is given to us we need to find what the maximum current that can be measured so what i'll do is i'll first calculate what the current sensitivity is and let's see what that number tells us and from that let's see if we can figure this out so let's calculate the current sensitivity the current sensitivity phi divided by i happens to be number of turns which is 500 times the coil area which is 3 times 10 to the power -4 and everything is in si units we need to take care of the units times the magnetic field which is 0.01 divided by c which is the spring constant 10 to the power minus phi and so what will i get if i put all of this well let's see the zeros and the zeros cancel over here i mean the decimals cancel i get five five threes are 15 and this will give me 110 on the top and so i get 150 so i get 150 and what is units degrees per ampere so i get degrees per ampere you might have a question how do we know it's a degrees or radians right that's a good question well here notice in the spring constant it's given 10 to the power minus 5 newton meters per degree so that tells me that this is in degrees okay so what is the meaning of this this means in our galvanometer if i send one amp of current the deflection we get is 150 degrees that's a pretty large deflection but now we are given that the maximum deflection that the galvanometer can accommodate is only 30. it cannot accommodate more than that so what's the max current so clearly if it can only accommodate 30 it will not be able to will not be able to send one amp of current through it so the question then is what is the maximum current that we can send and i think we can just do this logically i know that if i send one amp of current one amp of current i get a deflection of 150 degrees but i know that the maximum deflection i can get is 30 degrees so for 30 degrees how many amps of current i can send and remember the only reason i can do that is because we our galvanometer is linear it's a linear device and that's why i can do this cross multiplication and so if i do that i get what do i get i get the maximum current to be the maximum current to be 30 divided by 150 and that's one over five and that's 0.2 amperes so our galvanometer can only handle a max of 0.2 amperes of current so we can say that the range of our galvanometer is from 0 to 0.2 amperes so i can change my sticker now and put amperes and then this becomes an ammeter which can measure between 0 to 0.2 amperes