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Chapter 27: Problem 6

In which direction does a magnetic force act on an electron that is moving in the positive \(x\) -direction in a magnetic field pointing in the positive \(z\) -direction? a) the positive \(y\) -direction b) the negative \(y\) -direction c) the negative \(x\) -direction d) any direction in the \(x y\) -plane

Short Answer

Expert verified
Answer: b) the negative y-direction. Explanation: Using the Lorentz force equation and the right-hand rule, the magnetic force acts in the positive y-direction for a positive charge, but since the electron has a negative charge, the force is in the opposite direction, i.e., the negative y-direction.

Step by step solution

01

Determine the electron's charge

An electron has a negative charge, which can be represented as \(q = -e\), where \(e\) is the elementary charge (\(1.6 \times 10^{-19}\) Coulombs).
02

Write down the known quantities/vector components

An electron is moving in the positive \(x\)-direction, so its velocity vector is \(\vec{v} = v_x\hat{i}\), where \(v_x > 0\). The magnetic field is in the positive \(z\)-direction, so the magnetic field vector is \(\vec{B} = B_z\hat{k}\), where \(B_z > 0\).
03

Compute the cross product

The magnetic force acting on the electron is given by the Lorentz force equation: \[ \vec{F} = q(\vec{v} \times \vec{B}) \] Since \(q=-e\), we have: \[ \vec{F} = -e(\vec{v} \times \vec{B}) \] To take the cross product, we can use the determinant formula: \[ \vec{F} = -e\begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ v_x & 0 & 0 \\ 0 & 0 & B_z \end{vmatrix} \] Applying the determinant formula, we find: \[ \vec{F} = -e(\hat{i}(0 - 0) - \hat{j}(0 - 0) + \hat{k}(0 - 0)) = -e\cdot 0 \] So, the magnetic force is zero in the \(x\) and \(z\) directions.
04

Find the direction of the magnetic force in the \(y\)-direction

Now we need to use the right-hand rule to find the direction of the magnetic force in the \(y\)-direction. Point your right-hand fingers in the direction of the electron's velocity (\(+x\)), and then curl them toward the direction of the magnetic field (\(+z\)). Your thumb will be pointing in the direction of the force. Since the electron has a negative charge, the actual force will be in the opposite direction. In this case, your thumb will be pointing in the positive \(y\)-direction, but since the electron has a negative charge, the force is in the negative \(y\)-direction. Therefore, the correct answer is: b) the negative \(y\)-direction

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Most popular questions from this chapter

A particle with a charge of \(20.0 \mu \mathrm{C}\) moves along the \(x\) -axis with a speed of \(50.0 \mathrm{~m} / \mathrm{s}\). It enters a magnetic field given by \(\vec{B}=0.300 \hat{y}+0.700 \hat{z},\) in teslas. Determine the magnitude and the direction of the magnetic force on the particle.

The figure shows a schematic diagram of a simple mass spectrometer, consisting of a velocity selector and a particle detector and being used to separate singly ionized atoms \(\left(q=+e=1.60 \cdot 10^{-19} \mathrm{C}\right)\) of gold \((\mathrm{Au})\) and molybdenum (Mo). The electric field inside the velocity selector has magnitude \(E=1.789 \cdot 10^{4} \mathrm{~V} / \mathrm{m}\) and points toward the top of the page, and the magnetic field has magnitude \(B_{1}=1.00 \mathrm{~T}\) and points out of the page. a) Draw the electric force vector, \(\vec{F}_{E},\) and the magnetic force vector, \(\vec{F}_{B},\) acting on the ions inside the velocity selector. b) Calculate the velocity, \(v_{0}\), of the ions that make it through the velocity selector (those that travel in a straight line). Does \(v_{0}\) depend on the type of ion (gold versus molybdenum), or is it the same for both types of ions? c) Write the equation for the radius of the semicircular path of an ion in the particle detector: \(R=R\left(m, v_{0}, q, B_{2}\right)\). d) The gold ions (represented by the black circles) exit the particle detector at a distance \(d_{2}=40.00 \mathrm{~cm}\) from the entrance slit, while the molybdenum ions (represented by the gray circles) exit the particle detector at a distance \(d_{1}=19.81 \mathrm{~cm}\) from the entrance slit. The mass of a gold ion is \(m_{\text {gold }}=\) \(3.27 \cdot 10^{-25}\) kg. Calculate the mass of a molybdenum ion.

A proton with an initial velocity given by \((1.0 \hat{x}+$$2.0 \hat{y}+3.0 \hat{z})\left(10^{5} \mathrm{~m} / \mathrm{s}\right)\) enters a magnetic field given by \((0.50 \mathrm{~T}) \hat{z}\). Describe the motion of the proton

A magnetic field is oriented in a certain direction in a horizontal plane. An electron moves in a certain direction in the horizontal plane. For this situation, there a) is one possible direction for the magnetic force on the electron. b) are two possible directions for the magnetic force on the electron. c) are infinite possible directions for the magnetic force on the electron.

A copper wire with density \(\rho=8960 \mathrm{~kg} / \mathrm{m}^{3}\) is formed into a circular loop of radius \(50.0 \mathrm{~cm} .\) The cross-sectional area of the wire is \(1.00 \cdot 10^{-5} \mathrm{~m}^{2},\) and a potential difference of \(0.012 \mathrm{~V}\) is applied to the wire. What is the maximum angular acceleration of the loop when it is placed in a magnetic field of magnitude \(0.25 \mathrm{~T}\) ? The loop rotates about an axis through a diameter.
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