Ch 25: The Electric Potential

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 25: The Electric Potential

Problem 56

Chapter 25, Problem 56

A proton is fired from far away toward the nucleus of an iron atom. Iron is element number 26, and the diameter of the nucleus is 9.0 fm. What initial speed does the proton need to just reach the surface of the nucleus? Assume the nucleus remains at rest.

Verified step by step guidance

Identify the key concept: This problem involves the conservation of energy. The proton's initial kinetic energy is converted into electric potential energy as it approaches the nucleus. The electric potential energy is due to the Coulomb force between the positively charged proton and the positively charged nucleus.

Write the conservation of energy equation: The total energy is conserved, so the initial kinetic energy of the proton is equal to the electric potential energy at the surface of the nucleus. Mathematically, this is expressed as:

K_{i} = U_{f}

, where

K_{i} = (1/2)mv²

and

U_{f} = k(q₁q₂)/r

Substitute the known values into the potential energy formula: The charge of the proton is

q₁ = e

, and the charge of the iron nucleus is

q₂ = 26e

(since iron has 26 protons). The distance

r

is the radius of the nucleus, which is half the diameter:

= 9.0/2 \, \(\text{fm}\)

. The Coulomb constant is

= 8.99 \(\times\) 10^9 \, \(\text{N·m²/C²}\)

Set up the equation for the initial speed: Equating the initial kinetic energy to the electric potential energy, we get:

(1/2)mv² = k(q₁q₂)/r

. Solve for

v

= \(\sqrt{(2k(q₁q₂)/mr)}\)

, where

m

is the mass of the proton (

1.67 \(\times\) 10^{-27} \, \(\text{kg}\)

Substitute all numerical values into the equation: Use the given values for

k

q₁

q₂

r

, and

m

to calculate the initial speed

v

. Ensure all units are consistent (e.g., convert femtometers to meters:

1 \, \(\text{fm}\) = 10^{-15} \, \(\text{m}\)

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Coulomb's Law

Coulomb's Law describes the electrostatic force between charged particles. It states that the force is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them. In this scenario, the proton experiences a repulsive force from the positively charged nucleus of the iron atom as it approaches, which is crucial for determining the initial speed required to overcome this force.

Kinetic and Potential Energy

The concepts of kinetic and potential energy are fundamental in understanding the motion of the proton. Kinetic energy is the energy of motion, given by the formula KE = 1/2 mv², where m is mass and v is velocity. As the proton approaches the nucleus, its kinetic energy converts into electric potential energy due to the electrostatic interaction, which can be analyzed to find the initial speed needed to reach the nucleus.

Conservation of Energy

The principle of conservation of energy states that the total energy in a closed system remains constant. In this context, the initial kinetic energy of the proton must equal the potential energy at the point of closest approach to the nucleus. By applying this principle, one can calculate the necessary initial speed of the proton to ensure it has just enough energy to reach the surface of the nucleus without being repelled back.

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