# 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 600-๐F 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?

Weโll call the speed at which the bullet is traveling, 500 meters per second, ๐ฃ. And the acceptable distance of blurred motion, 1.00 millimeters, weโll call ๐. Weโre told we have a capacitor with a value of 600 microfarads. Weโll name that value capital ๐ถ. We want to solve for the resistance in the flash tube, which weโll call capital ๐.

If we start off drawing a sketch of our bulletโs motion as our bullet travels ahead with a speed ๐ฃ, weโre 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 ๐ถ, which will allow for a flash whose time constant guarantees that ๐ is the maximum amount of motion blur we experience.

The time constant ๐ก 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 ๐ก in terms of the motion of our bullet. And to do this, we can recall that an objectโs average speed ๐ฃ is equal to the distance it travels over the time it takes it to travel that distance.

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

When we solve for ๐, we find itโs equal to the acceptable blurred distance of the bullet ๐ divided by its speed ๐ฃ multiplied by the capacitance in the circuit ๐ถ. All three of these values are given in our problem statement. So weโre now ready to plug in and solve for ๐.

When we do plug in, weโre 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. Thatโs the resistance weโd need in our circuit in order to be able to photograph this bullet without unacceptable blur.
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