Two protons, starting several meters apart, are aimed directly at each other with speeds of 2.00×1052.00\$$times10^5 m/s$$, measured relative to the earth. Find the maximum electric force that these protons will exert on each other.

Ch 23: Electric Potential

Young & Freedman Calc - University Physics 14th Edition

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Young & Freedman Calc 14th Edition

Ch 23: Electric Potential

Problem 7

Chapter 23, Problem 7

Two protons, starting several meters apart, are aimed directly at each other with speeds of $2.00 \times 10^{5}$ m/s, measured relative to the earth. Find the maximum electric force that these protons will exert on each other.

Verified step by step guidance

Identify the relevant physical principles: The problem involves the electric force between two protons, which can be calculated using Coulomb's Law. The protons are initially moving towards each other, and we need to find the maximum force when they are closest.

Understand the concept of relative motion: Since both protons are moving towards each other with the same speed, their relative speed is the sum of their individual speeds. However, for calculating the force, we are interested in the point where they are closest, which is when their kinetic energy is completely converted into electric potential energy.

Apply conservation of energy: Initially, the protons have kinetic energy and negligible potential energy due to their large separation. As they approach each other, their kinetic energy decreases while their potential energy increases. At the point of closest approach, all kinetic energy is converted into electric potential energy.

Use the formula for electric potential energy: The electric potential energy (U) between two point charges is given by the formula:

\frac{k q_{1} q_{2}}{r}

, where k is Coulomb's constant, q1 and q2 are the charges of the protons, and r is the separation distance at closest approach.

Calculate the maximum electric force: The maximum electric force occurs at the point of closest approach and is given by Coulomb's Law:

F = \frac{k q^{2}}{r^{2}}

. Use the conservation of energy to find r, and then substitute it into this formula to find the maximum force.

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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 interaction between charged particles. It states that the force between two point charges is directly proportional to the product of their charges and inversely proportional to the square of the distance between them. The force is attractive if the charges are opposite and repulsive if they are the same.

Kinetic Energy and Relative Motion

Kinetic energy is the energy possessed by an object due to its motion, calculated as 1/2 mv^2, where m is mass and v is velocity. In this scenario, understanding relative motion is crucial as it affects how the protons approach each other, influencing the distance at which the maximum force occurs.

Electric Force and Maximum Interaction

The maximum electric force between two protons occurs when they are closest to each other, as per Coulomb's Law. As they approach, their kinetic energy is converted into potential energy, reducing their speed until they reach the point of closest approach, where the electric force is maximized.

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Work due to Electric Force

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Textbook Question

A particle with charge $positive 4.20$ nC is in a uniform electric field $E$ directed to the left. The charge is released from rest and moves to the left; after it has moved $6.00$ cm, its kinetic energy is $+ 2.20 x 10^{- 6}$ J. What is the work done by the electric force?

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Textbook Question

Two protons are released from rest when they are $0.750$ nm apart. What is the maximum acceleration they will achieve and when does this acceleration occur?

A small particle has charge $negative 5.00$ $μ$ C and mass $2.00 \times 10^{- 4}$ kg. It moves from point $A$ , where the electric potential is $V sub A equals positive 200$ V, to point $B$ , where the electric potential is $V sub B equals positive 800$ V. The electric force is the only force acting on the particle. The particle has speed $5.00$ m/s at point $A$ . What is its speed at point $B$ ? Is it moving faster or slower at $B$ than at $A$ ? Explain.

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Textbook Question

How much work would it take to push two protons very slowly from a separation of $2.00 \times 10^{- 10}$ m (a typical atomic distance) to $3.00 \times 10^{- 15}$ m (a typical nuclear distance)? If the protons are both released from rest at the closer distance in part (a), how fast are they moving when they reach their original separation?

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Textbook Question

A small metal sphere, carrying a net charge of $q_{1} = - 2.80$ μC, is held in a stationary position by insulating supports. A second small metal sphere, with a net charge of $q_{2} = - 7.80$ μC and mass $1.50$ g, is projected toward $q sub 1$ . When the two spheres are $0.800$ m apart, $q sub 2$ , is moving toward $q sub 1$ with speed $22.0$ m/s (Fig. E $23.5$ ). Assume that the two spheres can be treated as point charges. You can ignore the force of gravity. What is the speed of $q sub 2$ when the spheres are $0.400$ m apart?

A point charge $q sub 1 equals positive 2.40$ $μ$ C is held stationary at the origin. A second point charge $q_{2} = - 4.30$ $μ$ C moves from the point $x equals 0.150$ m, $y equals 0$ to the point $x equals 0.250$ m, $y equals 0.250$ m. How much work is done by the electric force on $q sub 2$ ?

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Two protons, starting several meters apart, are aimed directly at each other with speeds of 2.00×1052.00\(\times\)10^5 m/s, measured relative to the earth. Find the maximum electric force that these protons will exert on each other.

Verified video answer for a similar problem:

Key Concepts

Coulomb's Law

Kinetic Energy and Relative Motion

Electric Force and Maximum Interaction

Two protons, starting several meters apart, are aimed directly at each other with speeds of $2.00 \times 10^{5}$ m/s, measured relative to the earth. Find the maximum electric force that these protons will exert on each other.