A 3.350-m -long cylindrical capacitor consists of a solid...

A $3.350-\mathrm{m}$ -long cylindrical capacitor consists of a solid conducting core with a radius of 1.20 $\mathrm{mm}$ and an outer hollow conducting tube with an inner radius of 2.00 $\mathrm{mm}$ . The two conductors are separated by air and charged to a potential difference of 6.00 $\mathrm{V}$ . Calculate (a) the charge per length for the capacitor; (b) the total charge on the capacitor; (c) the capacitance; (d) the energy stored in the capacitor when fully charged.

Key Concepts

Cylindrical Capacitor

A cylindrical capacitor is composed of two coaxial cylindrical conductors separated by an insulating material. Its geometry leads to specific formulas for capacitance and charge distribution, differentiating it from parallel-plate or spherical capacitors. Understanding its configuration is key to analyzing its behavior under applied potential differences.

Capacitance Calculation

Capacitance is a measure of a capacitor's ability to store charge per unit voltage. In cylindrical capacitors, the capacitance is derived from the geometry of the conductors and the properties of the dielectric between them, utilizing logarithmic functions. This concept is essential in determining how the physical dimensions and material properties influence stored electrical energy.

Charge Distribution in Capacitors

The relationship between charge, voltage, and capacitance in a capacitor is defined by the equation Q = CV, where Q is the charge, C is the capacitance, and V is the voltage. In the context of cylindrical capacitors, this relationship is extended to consider charge per unit length, emphasizing how geometry affects charge accumulation along the length of the capacitor.

Energy Storage in Capacitors

The energy stored in a capacitor is given by formulas such as U = ½CV² or U = Q²/(2C), representing the work done to charge the capacitor. This concept connects the physical properties of the capacitor and its stored electrical energy, and it is fundamental when evaluating the performance and efficiency of capacitors in electrical circuits.

Transcript

00:01 Hi, everybody.

00:02 So we need to find a, the charge per length of capsidors.

00:06 So charge per length, we have charge per length equals cab times b -a -b divided by length.

00:17 So we have two pi - epsilon not divided by l -n -r -b -r -a times velocity ab.

00:29 And this is going to equal to 2 pi 8 .8 .5 times 10 to negative 12.

00:43 F, there is over meters times 6 volts divided by you have your ln as 2mm over 1 .2m.

01:04 I'm just going to like separate this here.

01:07 And if you're curious, this part is, so we are looking at this part here, and that's where we're getting these, okay? let me just go back and erase these, so that way you don't get confused.

01:24 Okay? so this is going to equal to equivalent calculator 60, 6654 point.

01:34 Make that four look nice.

01:37 I'm just not good at making fours, i realize.

01:41 64 .275 p .c over m.

01:49 Okay.

01:51 And now for b, we have, let me get down, to pull that all down here.

01:57 And b is a total charge on capsiders.

02:00 So we have the charge here.

02:03 It's going to equal charge, per length divided times l.

02:09 So we have 6 .54 .275 p .c over m times 0 .335 meters.

02:26 And this is going to give us 2 .98 .99 p...