Problem 55 - University Physics with Modern Physics

Question

When a particle of charge \(q > 0\) moves with a velocity of \(\vec { \boldsymbol { v } } _ { 1 }\) at \(45.0 ^ { \circ }\) from the \(\pm x\) -axis in the \(x y\) -plane, a uniform magnetic field exerts a force \(F _ { 1 }\) along the \(- z\) -axis (Fig. \(P 27.55 ) .\) When the same particle moves with a velocity \(\vec { \boldsymbol { v } } _ { 2 }\) with the same magnitude as \(\vec { \boldsymbol { v } } _ { 1 }\) but along the \(+ z\) -zaxis, a force \(\vec { \boldsymbol { F } } _ { 2 }\) of magnitude \(F _ { 2 }\) is exerted on it along the \(+ x\) -axis. (a) What are the magnitude (in terms of \(q , v _ { 1 } ,\) and \(F _ { 2 }\) ) and direction of the magnetic field? (b) What is the magnitude of \(\vec { F } _ { 1 }\) in terms of \(F _ { 2 } ?\)

Step-by-Step Solution

Answer

(a) \(B = \frac{F_2}{q v_1} \hat{j}\), (b) \(F_1 = \frac{F_2}{\sqrt{2}}\).

Step 1: Analyze the Force Equation

The magnetic force on a charged particle moving in a magnetic field is given by the equation \[\vec{F} = q \vec{v} \times \vec{B}\]where \(\vec{F}\) is the force, \(q\) is the charge, \(\vec{v}\) is the velocity, and \(\vec{B}\) is the magnetic field. The cross product indicates the force is perpendicular to both velocity and magnetic field.

Step 2: Resolve Velocity Vectors

For motion in the \(xy\)-plane at an angle of \(45.0^{\circ}\) along the \(\pm x\)-axis, the components of \(\vec{v}_1\) can be expressed as:\[\vec{v}_1 = v_1 \left( \frac{1}{\sqrt{2}} \hat{i} + \frac{1}{\sqrt{2}} \hat{j} \right)\]For motion along the \(+z\)-axis, \(\vec{v}_2\) is:\[\vec{v}_2 = v_1 \hat{k}\]

Step 3: Analyze Forces and Apply Right-Hand Rule

For \(\vec{v}_1\), the force \(\vec{F}_1\) is along the \(-z\)-axis, suggesting that \(\vec{B}\) lies in the \(xy\)-plane. Using the cross product, the equation becomes:\[\vec{F}_1 = q \vec{v}_1 \times \vec{B} = -F_1 \hat{k}\]For \(\vec{v}_2\), the force \(\vec{F}_2\) is along \(+x\)-axis, and hence,\[\vec{F}_2 = q \vec{v}_2 \times \vec{B} = F_2 \hat{i}\]

Step 4: Determine the Direction of Magnetic Field

From \(\vec{F}_2 = q v_1 B_2 \hat{i}\), where \(B_2\) is the component of \(\vec{B}\) in the \(-y\)-direction, it implies:\[F_2 = q v_1 B_2 \rightarrow B_2 = \frac{F_2}{q v_1}\]Similarly, from \(\vec{F}_1 = q v_1 B_1 \hat{k} \), directed along \(-z\):\[B_1 = B \cos 45^{\circ} = \frac{F_2}{\sqrt{2} q v_1}\]Hence, \(\vec{B} = \frac{F_2}{q v_1} \hat{j}\).

Step 5: Calculate the Magnitude of \(\vec{F}_1\)

Using the known relation for force:\[F_1 = q v_1 B \, (\cos 45^{\circ})\]Since the magnetic field has magnitude:\[B = \frac{F_2}{q v_1},\]we find:\[F_1 = \frac{F_2}{\sqrt{2}}\]

Key Concepts

Cross Product

The cross product is a crucial mathematical tool when dealing with magnetic forces on charged particles. It allows us to determine the direction and magnitude of the force exerted by the magnetic field. When a charged particle moves through a magnetic field, the force is calculated using the equation \(\vec{F} = q \vec{v} \times \vec{B}\). Here, \(q\) is the charge of the particle, \(\vec{v}\) is the velocity vector, and \(\vec{B}\) represents the magnetic field vector. The cross product \(\times\) results in a vector that is perpendicular to both \(\vec{v}\) and \(\vec{B}\). This means that the resulting force vector \(\vec{F}\) does not lie along the direction of motion or the magnetic field but forms a right-angle with both. This unique property is essential for understanding the motion of charged particles in electromagnetic fields.

Velocity Vectors

Velocity vectors describe the speed and direction of particles in motion. In the context of magnetic forces, understanding these vectors is crucial. In the exercise, the particle moves with a velocity \(\vec{v}_1\) at \(45^{\circ}\) from the \(\pm x\)-axis in the \(xy\)-plane. This can be expressed as \(\vec{v}_1 = v_1 \left( \frac{1}{\sqrt{2}} \hat{i} + \frac{1}{\sqrt{2}} \hat{j} \right)\). Each component of this vector represents the velocity in the \(x\) and \(y\) directions, respectively.

Also, when the particle moves along the \(+z\)-axis, its velocity vector changes to \(\vec{v}_2 = v_1 \hat{k}\). Here, \(\hat{i}, \hat{j},\) and \(\hat{k}\) are the unit vectors along the \(x\), \(y\), and \(z\) axes, respectively. This decomposition of the velocity vectors enables us to apply the cross product and analyze the specific components of the magnetic force acting in different directions.

Right-Hand Rule

The right-hand rule is a simple method to determine the direction of the resulting vector in a cross-product operation. It is particularly useful in physics when dealing with forces in three-dimensional space. To apply the right-hand rule, you align your right hand with the first vector \(\vec{v}\) and then curl your fingers towards the second vector \(\vec{B}\). Your thumb will point in the direction of the resulting force \(\vec{F}\).

For instance, in the exercise, when using \(\vec{v}_1\) and the unknown magnetic field \(\vec{B}\), the force \(\vec{F}_1\) is observed along the \(-z\)-axis. Employing the right-hand rule consistently helps to ensure the accuracy of the magnetic force's direction and is a valuable tool for visualizing the effects of cross products in electromagnetic problems.

Magnetic Field Direction

In this exercise, determining the direction of the magnetic field involves analyzing how it influences the force experienced by a moving particle. The magnetic field vector \(\vec{B}\) affects the direction of \(\vec{F}_1\) and \(\vec{F}_2\) as the particle moves with different velocity vectors \(\vec{v}_1\) and \(\vec{v}_2\).

Given that \(\vec{F}_1\) acts along the \(-z\)-axis when \(\vec{v}_1\) is considered, it is implied that the magnetic field lies in the \(xy\)-plane. Specifically, working through the crossed-product relations and the force direction, the magnetic field is determined to have its components as \(\vec{B} = \frac{F_2}{q v_1} \hat{j}\), mainly directed along the \(y\)-axis.

This understanding shows how the magnetic field’s orientation directly affects the path and force on the charged particles in motion, a fundamental principle in electromagnetism.