91Ó°ÊÓ

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.)

Step by step solution

01

Estimation of Hair Length (L) and Mass (m)

A typical human hair is about 18 centimeters long (\(L = 18 cm = 0.18 m\)). Using the average linear mass density of hair (\(65µg/cm\)) we estimate the mass of hair as \(m = density \times length = 65µg/cm \times 18cm = 1170µg = 1.17 \times 10^{-6} kg\).

02

Estimation of Number of Hairs (N)

On any given day, after combing, there could be approximately 100 strands of hair that stands out due to electrostatic repulsion. So, we take \(N = 100\).

03

Solving for charge (Q)

Given that the electrostatic force equals twice the weight of hair (\(2mg\)), we can substitute the values of m, g (acceleration due to gravity), and N into the given equation to solve for Q. Rearranging gives us, \(Q = \sqrt{(2mg \times 4πε_0 \times L^2) / N}\).

04

Calculation of Head's Capacitance

Assuming your head as a sphere with typical adult head radius of approximately 9cm (\(R = 9cm = 0.09m\)). Substituting this into the capacitance equation gives us \(C = 4πε_0R\).

05

Calculation of potential (V)

Potential is calculated by dividing the charge by the capacitance, \(V = Q/C\). Substitute the calculated value of Q and C to get the potential.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Electrostatic Repulsion

When you comb your hair and your strands stand on end, it's a fascinating display of electrostatic repulsion in action. Electrostatic repulsion occurs when two objects with like charges interact, creating a force that pushes them apart. In the case of your hair, it stands up because each strand gains a similar electric charge that repels against the others. This is due to the fact that objects with the same charge, whether positive or positive or negative or negative, will experience a repulsive force.

For students trying to visualize this, consider that each hair becomes a tiny magnet with the same pole facing each other; naturally, they want to distance themselves, thus standing up. Understanding how these forces work explains a wide range of phenomena from the mundane, like why certain materials cling together when rubbed, to the essential, such as how electrons are held in atoms.

Linear Mass Density

Linear mass density is a measure of mass per unit length, a concept vital to understanding various physical phenomena, including the exercise at hand. It's typically used in situations where an object's mass is distributed along a line, such as a string, a rod, or in our case, a strand of hair. Understanding linear mass density helps in estimating not only the mass of an object given its length but also in calculations involving tension, waves on strings, and in this scenario, the balance of electrostatic and gravitational forces.

For example, to determine the electrostatic force needed to counteract gravity and raise a hair strand, knowing the mass of the hair via its linear mass density is crucial. This parameter allows us to bridge the microscopic world of charges with the macroscopic world we observe, as seen when we calculate the force needed to lift your hair.

Capacitance

Capacitance is a measure of an object’s ability to store charge, an essential concept in both physics and engineering. In our hair-raising scenario, thinking of your head as a spherical capacitor can provide insights into how much charge it can hold and how that relates to the electric potential it can reach.

The formula for the capacitance of a sphere, C = 4πε₀R, allows us to estimate how much electrostatic energy your head accumulates during combing. Capacitance does not just relate to static electricity; it's also a cornerstone in understanding how circuits work and how energy is stored in different components. For example in capacitors on electronic boards, it plays a fundamental role in timing circuits, filters, and energy storage systems.

Electric Potential

Electric potential, a key concept when discussing electrostatic phenomena, is the amount of electric potential energy per unit charge at a particular position in an electric field. When we calculate the potential your head achieves after being charged by a comb, we're really measuring the capacity of the electric field generated by your head to do work on a charged particle. This potential is what would drive an electric current if a conductive path were provided.

A higher electric potential indicates a stronger electric field, which translates to a greater ability to move charges and create electrostatic forces. In the context of our exercise, knowing the potential helps explain not only how your hair lifts but also how other objects might be influenced by the static charge on your head, showcasing how electric potential links to the forces experienced by charged objects.