Video: Finding the Resistance of an RC Circuit given Its Capacitance and Its Time Constant

If you wish to take a picture of a bullet traveling at 500 m/s, then a very brief flash of light produced by an RC discharge can limit blurring. Assuming 1.00 mm of motion during one RC constant is acceptable, and given that the flash is driven by a 600-๐œ‡F capacitor, what is the resistance in the flash tube?

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Video Transcript

If you wish to take a picture of a bullet traveling at 500 meters per second, then a very brief flash of light produced by an RC discharge can limit blurring. Assuming 1.00 millimeters of motion during one RC constant is acceptable, and given that the flash is driven by a 600-microfarad capacitor, what is the resistance in the flash tube?

Weโ€™ll call the speed at which the bullet is traveling, 500 meters per second, ๐‘ฃ. And the acceptable distance of blurred motion, 1.00 millimeters, weโ€™ll call ๐‘‘. Weโ€™re told we have a capacitor with a value of 600 microfarads. Weโ€™ll name that value capital ๐ถ. We want to solve for the resistance in the flash tube, which weโ€™ll call capital ๐‘….

If we start off drawing a sketch of our bulletโ€™s motion as our bullet travels ahead with a speed ๐‘ฃ, weโ€™re told that when we photograph it, an acceptable amount of blurring involves the distance ๐‘‘ given as 1.00 millimeters. We want to solve for the resistance of the circuit including a capacitor ๐ถ, which will allow for a flash whose time constant guarantees that ๐‘‘ is the maximum amount of motion blur we experience.

The time constant ๐‘ก in an RC circuit is equal simply to the product of the resistance and the capacitance in that circuit. We also want to be able to express time ๐‘ก in terms of the motion of our bullet. And to do this, we can recall that an objectโ€™s average speed ๐‘ฃ is equal to the distance it travels over the time it takes it to travel that distance.

So ๐‘ฃ equals ๐‘‘ over ๐‘ก or ๐‘ก equals ๐‘‘ over ๐‘ฃ. In our case, we can also write that this equation is equal to ๐‘… times ๐ถ because weโ€™ve set the condition that our time can be no more than one time constant of our circuit. So weโ€™d like to use this new equation to solve for the resistance in our circuit ๐‘….

When we solve for ๐‘…, we find itโ€™s equal to the acceptable blurred distance of the bullet ๐‘‘ divided by its speed ๐‘ฃ multiplied by the capacitance in the circuit ๐ถ. All three of these values are given in our problem statement. So weโ€™re now ready to plug in and solve for ๐‘….

When we do plug in, weโ€™re careful to write our distance in units of meters and our capacitance in units of farads. When we enter these values on our calculator, we find that, to three significant figures, ๐‘… equals 3.33 times 10 to the negative third ohms. Thatโ€™s the resistance weโ€™d need in our circuit in order to be able to photograph this bullet without unacceptable blur.

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