Ch 23: The Electric Field

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796당신이 사용하는 게 아니라요?교과서 변경

Knight Calc 5th Edition

Ch 23: The Electric Field

문제 73

23장, 문제 73

A proton orbits a long charged wire, making $1.0 \times 10^{6}$ revolutions per second. The radius of the orbit is $1.0$ cm. What is the wire's linear charge density?

검증된 단계별 안내

Start by identifying the force acting on the proton. The centripetal force required to keep the proton in circular motion is provided by the electrostatic force between the proton and the charged wire. The centripetal force is given by:

F

_c = mv²/r, where

m

is the mass of the proton,

v

is its velocity, and

r

is the radius of the orbit.

The electrostatic force between the proton and the charged wire is given by:

F

_e = qE, where

q

is the charge of the proton and

E

is the electric field due to the charged wire. For a long charged wire, the electric field at a distance

r

is given by:

E = 2 k λ / r

, where

λ

is the linear charge density and

k

is Coulomb's constant.

Equate the centripetal force to the electrostatic force:

m v

²/r = q(2kλ/r). Simplify the equation to solve for

λ

λ = (m v

²)/(2kq).

Determine the velocity of the proton. The proton completes

1.0 × 10

⁶ revolutions per second, so its angular velocity is

ω = 2 π (1.0 × 10

⁶). The linear velocity is related to the angular velocity by

v = ω r

. Substitute the radius

r = 0.01

m to find

v

Substitute the known values into the expression for

λ

. Use

m = 1.67 × 10

^-27 kg (mass of a proton),

q = 1.6 × 10

^-19 C (charge of a proton),

k = 8.99 × 10

⁹ N·m²/C², and the calculated value of

v

. Solve for

λ

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

14m

도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Centripetal Force

Centripetal force is the net force acting on an object moving in a circular path, directed towards the center of the circle. For a charged particle like a proton orbiting a charged wire, this force is provided by the electric field created by the wire. The balance between the centripetal force required for circular motion and the electric force due to the wire's charge is crucial for determining the wire's charge density.

Electric Field and Force

The electric field around a charged object exerts a force on other charges within that field. For a long charged wire, the electric field can be calculated using Gauss's law, which relates the electric field to the linear charge density. The force experienced by the proton due to this electric field is what keeps it in circular motion, allowing us to relate the charge density to the proton's orbit.

Linear Charge Density

Linear charge density is defined as the amount of electric charge per unit length along a charged object, such as a wire. It is denoted by the symbol λ (lambda) and is measured in coulombs per meter (C/m). In this problem, calculating the linear charge density of the wire involves using the relationship between the electric field produced by the wire and the centripetal force acting on the proton in its circular orbit.

관련 실천

교과서 질문

A rod of length L lies along the y-axis with its center at the origin. The rod has a nonuniform linear charge density $λ = a ∣ y ∣$ , where a is a constant with the units C/m². Find the electric field strength of the rod at distance x on the x-axis.

110

views

교과서 질문

An infinitely long sheet of charge of width L lies in the xy-plane between x = -L /2 and x = L /2. The surface charge density is h. Draw a graph of field strength E versus x for x > L /2.

448

views

교과서 질문

One type of ink-jet printer, called an electrostatic ink-jet printer, forms the letters by using deflecting electrodes to steer charged ink drops up and down vertically as the ink jet sweeps horizontally across the page. The ink jet forms 30-μm-diameter drops of ink, charges them by spraying 800,000 electrons on the surface, and shoots them toward the page at a speed of 20 m/s . Along the way, the drops pass through two horizontal, parallel electrodes that are 6.0 mm long, 4.0 mm wide, and spaced 1.0 mm apart. The distance from the center of the electrodes to the paper is 2.0 cm. To form the tallest letters, which have a height of 6.0 mm, the drops need to be deflected upward (or downward) by 3.0 mm. What electric field strength is needed between the electrodes to achieve this deflection? Ink, which consists of dye particles suspended in alcohol, has a density of 800 kg/m³.

1681

views

rank

A proton orbits a long charged wire, making 1.0×1061.0×10^6 revolutions per second. The radius of the orbit is 1.01.0 cm. What is the wire's linear charge density?

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Centripetal Force

Electric Field and Force

Linear Charge Density

A proton orbits a long charged wire, making $1.0 \times 10^{6}$ revolutions per second. The radius of the orbit is $1.0$ cm. What is the wire's linear charge density?