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(I) A proton is released in a uniform electric field, and it experiences an electric force of \(2.18 \times 10^{-14} \mathrm{~N}\) toward the south. What are the magnitude and direction of the electric field?

Short Answer

Expert verified
Magnitude: \(1.363 \times 10^{5} \text{ N/C}\), Direction: South.

Step by step solution

01

Identify the Given Information

We know that a proton experiences an electric force of \(2.18 \times 10^{-14} \text{ N}\) toward the south in the electric field. We need to find the electric field's magnitude and direction.
02

Use the Formula for Electric Force

The electric force \(F\) on a charge \(q\) in an electric field \(E\) is given by the formula \(F = qE\). We need to rearrange this equation to find \(E\).
03

Rearrange the Formula to Solve for Electric Field

To find the electric field \(E\), rearrange the formula: \(E = \frac{F}{q}\). Here, \(F = 2.18 \times 10^{-14} \text{ N}\) and \(q = 1.60 \times 10^{-19} \text{ C}\), the charge of a proton.
04

Calculate the Electric Field Magnitude

Substitute the values into the equation: \(E = \frac{2.18 \times 10^{-14} \text{ N}}{1.60 \times 10^{-19} \text{ C}} = 1.3625 \times 10^{5} \text{ N/C}\). Therefore, the magnitude of the electric field is \(1.363 \times 10^{5} \text{ N/C}\) when rounded to three significant figures.
05

Determine the Direction of the Electric Field

The electric field direction is defined as the direction of the force on a positive charge (proton in this case). Since the force is toward the south, the electric field also points toward the south.

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Key Concepts

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

Electric Force
Electric force is the interaction that occurs between charged particles. It's a fundamental concept in physics and plays a vital role in understanding how electric fields affect charges. When a charge is placed in an electric field, it experiences a force due to the field's influence. This is known as the electric force. For example, in the exercise, a proton feels an electric force of \(2.18 \times 10^{-14} \text{ N}\) toward the south.
  • The electric force can be attractive or repulsive, depending on the nature of the charges.
  • A positively charged particle, like a proton, experiences a force in the same direction as the electric field.
The formula for electric force is given by \(F = qE\), where \(F\) is the force, \(q\) is the charge, and \(E\) is the electric field magnitude. Understanding this relationship helps to calculate how charges interact with electric fields.
Proton Charge
The proton charge is a fundamental quantity and is essential to calculate electric interactions. A proton is a subatomic particle found in the nucleus of an atom and has a positive charge. This charge is notably one of the constants used in physics, with a value of \(1.60 \times 10^{-19} \text{ C}\) (coulombs). Knowing the charge of a proton allows us to determine how it will behave in an electric field.
  • Being positively charged, protons move in the direction of the electric field if they are free to move.
  • The proton charge is equal in magnitude but opposite in sign to the electron's charge.
Understanding the proton charge is crucial when dealing with problems related to electric forces and fields, as it helps in accurately determining the electric force acting on it.
Field Magnitude
The field magnitude refers to the strength of the electric field at a given point and is usually measured in newtons per coulomb (\(\text{N/C}\)). In the context of our exercise, when a proton is subjected to an electric field and feels a force, this force is directly related to the field magnitude. To calculate the field magnitude, use the formula \(E = \frac{F}{q}\), where \(E\) is the electric field magnitude, \(F\) is the force experienced by the charge, and \(q\) is the charge of the particle (in this case, a proton). By substituting the known values:- Force \(F = 2.18 \times 10^{-14} \text{ N}\)- Charge \(q = 1.60 \times 10^{-19} \text{ C}\)You find that:\[E = \frac{2.18 \times 10^{-14} \text{ N}}{1.60 \times 10^{-19} \text{ C}} = 1.363 \times 10^{5} \text{ N/C}\]Thus, the field magnitude is \(1.363 \times 10^{5} \text{ N/C}\), providing an understanding of the field's strength at that point.
Field Direction
Field direction indicates the orientation of the electric field at a certain point. It's essential for predicting how a charged particle will move in the field. For positive charges, such as protons, the field direction is the same as the direction of the force exerted on the charge by the electric field. In the exercise example, since the proton experiences an electric force toward the south, it implies that the electric field direction is also toward the south.
  • Direction is significant because it helps determine the trajectory or path that a charged particle will follow within the field.
  • Knowing both the magnitude and direction of the field gives a complete picture of the electric field's characteristics.
The rule is simple: for positive charges like protons, they will move along the lines of the electric field as indicated by the direction of the force they experience.

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Most popular questions from this chapter

One type of electric quadrupole consists of two dipoles placed end to end with their negative charges (say) overlapping; that is, in the center is \(-2 Q\) flanked (on a line) by \(a+Q\) to either side (Fig. \(21-74) .\) Determine the electric field \(\overrightarrow{\mathbf{E}}\) at points along the perpendicular bisector and show that \(E\) decreases as \(1 / r^{4}\). Measure \(r\) from the \(-2 Q\) charge and assume \(r \gg \ell\)

(III) Suppose a uniformly charged wire starts at point 0 and rises vertically along the positive \(y\) axis to a length \(\ell\) . (a) Determine the components of the electric field \(E _ { x }\) and \(E _ { y }\) at point \(( x , 0 ) .\) That is, calculate \(\vec { \mathbf { E } }\) near one end of a long wire, in the plane perpendicular to the wire. (b) If the wire extends from \(y = 0\) to \(y = \infty ,\) so that \(\ell = \infty ,\) show that \(\vec { \mathbf { E } }\) makes a \(45 ^ { \circ }\) angle to the horizontal for any \(x .\) [Hint: See Example 11 of "Electric Charge and Electric Field" andFig. \(29 . ]\)

In a simple model of the hydrogen atom, the electron revolves in a circular orbit around the proton with a speed of \(2.2 \times 10 ^ { 6 } \mathrm { m } / \mathrm { s }\) . Determine the radius of the electron's orbit. [ Hint: Recall circular motion.

(III) A thin glass rod is a semicircle of radius \(R\), Fig. \(21-66\). A charge is nonuniformly distributed along the rod with a linear charge density given by \(\lambda=\lambda_{0} \sin \theta,\) where \(\lambda_{0}\) is a positive constant. Point \(\mathbf{P}\) is at the center of the semicircle. (a) Find the electric field \(\overrightarrow{\mathbf{E}}\) (magnitude and direction) at point P. [Hint: Remember \(\sin (-\theta)=-\sin \theta,\) so the two halves of the rod are oppositely charged.] (b) Determine the acceleration \(\quad\) (magnitude \(\quad\) and direction) of an electron placed at point \(\mathrm{P},\) assuming \(\quad R=1.0 \mathrm{~cm}\) and \(\lambda_{0}=1.0 \mu \mathrm{C} / \mathrm{m}\)

(II) Determine the magnitude of the acceleration experienced by an electron in an electric field of \(576 \mathrm{~N} / \mathrm{C}\). How does the direction of the acceleration depend on the direction of the field at that point?

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