After combing your hair on a dry day, some of your hair...

After combing your hair on a dry day, some of your hair stands up, forced away from your head by electrostatic repulsion. (a) Estimate the length $L$ of your hair. (b) Using the average linear mass density of hair, which is $65 \mu \mathrm{g} / \mathrm{cm},$ estimate the mass $m$ of one of your hairs. (c) Estimate the number $N$ of hairs that stand after combing. (d) Assume that the comb has taken away a charge $-2 Q,$ and that your hair has therefore gained an amount of charge $2 Q .$ Assume further that half of this charge resides next to your head and the other half is distributed at the ends of the $N$ strands that stand up. Assume the electrostatic force that lifted a hair was twice its weight. Show that this leads to $2 m g=\frac{1}{4 \pi \epsilon_{0}} \frac{Q^{2} / N}{L^{2}}$ Use this equation to estimate the charge $Q$ that resides on your head. (e) If your head were a sphere with radius $R$, it would have a capacitance of $4 \pi \epsilon_{0} R .$ Estimate the radius of your head; then use your estimate to determine your head's capacitance. (f) Use your result to estimate the potential attained due to combing. (The surprising result illustrates an interesting feature of static electricity.)

After combing your hair on a dry day, some of your hair stands up, forced away from your head by electrostatic repulsion.
(a) Estimate the length $L$ of your hair. (b) Using the average linear mass density of hair, which is $65 \mu \mathrm{g} / \mathrm{cm},$ estimate the mass $m$ of one of your hairs. (c) Estimate the number $N$ of hairs that stand after combing.
(d) Assume that the comb has taken away a charge $-2 Q,$ and that your hair has therefore gained an amount of charge $2 Q .$ Assume further that half of this charge resides next to your head and the other half is distributed at the ends of the $N$ strands that stand up. Assume the electrostatic force that lifted a hair was twice its weight. Show that this leads to
$2 m g=\frac{1}{4 \pi \epsilon_{0}} \frac{Q^{2} / N}{L^{2}}$ Use this equation to estimate the charge $Q$ that resides on your head.
(e) If your head were a sphere with radius $R$, it would have a capacitance of $4 \pi \epsilon_{0} R .$ Estimate the radius of your head; then use your estimate to determine your head's capacitance. (f) Use your result to estimate the potential attained due to combing. (The surprising result illustrates an interesting feature of static electricity.)

Key Concepts

Electric Potential

Electric potential represents the work done per unit charge to move a charge from a reference point to a specific point in an electric field. In the context of static electricity, calculating the potential of a charged object involves its capacitance and the charge it holds, illustrating how everyday actions can generate significant voltages.

Coulomb's Law

Coulomb's law quantifies the electrostatic force between two point charges. It states that the force is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. This principle is crucial for understanding how charged objects, such as strands of hair carrying charge, repel each other.

Electrostatic Force and Force Balance

In systems where both gravitational and electrostatic forces act, achieving equilibrium requires balancing these forces. For example, when the upward electrostatic repulsion on a charged hair strand equals twice its weight, the equilibrium condition can be used to estimate the amount of electric charge present.

Mass Density and Estimation Techniques

Mass density, particularly in a linear form, relates the mass of an object to its length. Using average linear mass density enables the estimation of the mass of objects from their dimensions. Estimation techniques involve approximating physical quantities from typical or average values, which is essential in problems where exact measurements are not available.

Capacitance of a Conductor

Capacitance is a measure of a conductor's ability to store electric charge per unit potential difference. For a spherical conductor, the capacitance is directly proportional to its radius. This concept connects the stored charge on an object to its potential and is fundamental in evaluating how combing can lead to unexpectedly high voltages.

Transcript

0:00 Hi there.

00:01 So for this problem, we are told that our hair after a dry day, and some of our hair stamps up, force away from our head by electrostatic repulse.

00:15 So the first thing that we need to do for this is to estimate the length of our hair.

00:24 So we need to estimate that.

00:26 And we are going to estimate that and the length.

00:32 The length or far of our hair is 15 centimeters.

00:37 Of course, this depends on the person that is going to estimate that value.

00:42 And this, if we want to transform it, if we want to transform it into meters, we divide that value by 100, so it will be 0 .15 meters.

00:56 Now, part b of this problem tell us that using the average linear mass density, of earth, which is going to put that in here, the density, well, linear density, we're going to call that just, what are kind of, we're going to call a row.

01:23 And that's 65 micrograms per centimeter.

01:33 So the problem sets that we need to estimate the mass of one of our hair.

01:44 So with that, we just simply multiply that value by the length of it.

01:53 That will give us the mass of one of our hair.

01:57 So we multiply 65 micrograms per centimeter times 15 centimeter.

02:06 That's the length of our hair.

02:10 So this will give us a value of 975 micrograms.

02:19 So that's the estimation for the mass of one hair.

02:25 Now, par c tell us to estimate the numbers of her that stand after combing.

02:34 So we're going to say just once again this is an estimation.

02:41 You could do this with whatever number you want, but i'm going to say that the number of hers that stand up after combing are 25 hertz.

02:56 Now, for part d of this problem, we are told to assume that the comb has taken away a charge of minus q.

03:06 Minus 2 times q.

03:09 So minus 2 times q.

03:14 And also told us that our hair has therefore gained an amount of charge of 2 q.

03:25 And so we are given in here to assume that half of this charge resides not to our hair.

03:33 And it is distributed at the ends of the end, strands and stand up.

03:39 So we need to assume the electrosity force that lifted a herd was twice is weight.

03:46 So we need to show that this is the what did you show that the themes that the problem set leads to this expression right here.

04:05 Q over and over the length square.

04:15 So it says to use this equation to estimate the chart that resides on our head.

04:26 And we first i'm going to start by remember that the electric force is given by 1 over 4 pi epsilon sub 0 or the vacuum permissivity times q1 and the product between q and the charges involved over the distance square.

04:52 So this is the force between the charges at both ends of our hair.

04:58 So we know that that will be q1 is equal to q2.

05:03 And we know that that is the charts for each hair.

05:08 So it will be q over the numbers of herd that we stipulate in here, that we just assume or make it.

05:17 Assumption.

05:20 And we also know that the distance from one point to the other, the distance are, is equal to the length of the herd.

05:33 And the force is twice the weight of the hair, so which is times the weight, which is equal to two times the mass of the herd times the acceleration due to gravity.

05:47 So we need just to simply equal to this force, this gravitational force to the electric force.

06:00 So we will have the following...

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