The arm of a moving-coil galvanometer is deflected through an angle of 22 degrees when the current through the galvanometer is 360 microamperes. The maximum deflection angle for the arm is 45 degrees. What is the maximum value of current that the galvanometer can measure? Answer to the nearest microampere.
To begin, let’s sketch a diagram. And although the entire galvanometer isn’t drawn here, the dashed lines imply that the system continues to form a complete circuit, allowing for charge flow. Let’s quickly recall that the galvanometer arm, or pointer, deflects through a larger angle 𝜃 when the magnitude of current in the galvanometer 𝐼 is greater. We can relate the arm deflection and amount of current by determining a galvanometer’s sensitivity, represented by 𝑆. The sensitivity equals 𝜃 divided by 𝐼. Thus, for a galvanometer with a low sensitivity, the arm will deflect less for a given current. Likewise, for a greater sensitivity, it won’t take as much current to make the arm deflect more.
Here, we know that when 𝐼 equals 360 microamps, 𝜃 equals 22 degrees. We could calculate the sensitivity of this galvanometer by substituting these values into the formula for 𝑆 so that we have 22 degrees over 360 microamps. Note that for a given galvanometer, its sensitivity has to remain constant to ensure it makes accurate measurements. This is important. The ratio of arm deflection to current, which equals sensitivity, must remain constant for all measurements made by this one galvanometer.
Now, for this galvanometer, we know that the maximum angle for arm deflection, which we’ll call 𝜃 max, is 45 degrees. And we want to find the maximum value of current, which we’ll call 𝐼 max, that this angle corresponds to. Since we’re talking about the same galvanometer here, we know that the value 𝜃 max divided by 𝐼 max must have the same value as the 𝜃 divided by 𝐼 that we wrote down earlier. So let’s set these quantities equal to each other. Now, we have 22 degrees over 360 microamps equals 45 degrees over 𝐼 max. To solve for 𝐼 max, let’s cross multiply. And we have 22 degrees times 𝐼 max equals 45 degrees times 360 microamps. To get 𝐼 max by itself, we divide both sides of the equation by 22 degrees. So that term cancels out of the left-hand side.
Notice that we can also cancel out units of degrees on the right-hand side, leaving only units of microamperes for the answer. Now, evaluating 45 times 360 divided by 22 gives us 736.36 recurring. And finally, rounding our answer to the nearest microampere, we’ve found that the galvanometer can measure a maximum current value of 736 microamperes.