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Magnetic fields are produced by moving charges and exert forces on moving charges. When a particle with charge \(q\) is moving with velocity \(\overrightarrow{\boldsymbol{v}}\) in a magnetic field \(\overrightarrow{\boldsymbol{B}},\) the force \(\overrightarrow{\boldsymbol{F}}\) that the field exerts on the particle is given by \(\overrightarrow{\boldsymbol{F}}=q \overrightarrow{\boldsymbol{v}} \times \overrightarrow{\boldsymbol{B}}\). The SI units are as follows: For charge it is the coulomb (C), for magnetic field it is tesla (T), for force it is newton \((\mathrm{N}),\) and for velocity it is \(\mathrm{m} / \mathrm{s} .\) If \(q=-8.00 \times 10^{-6} \mathrm{C}, \overrightarrow{\boldsymbol{v}}\) is \(3.00 \times 10^{4} \mathrm{~m} / \mathrm{s}\) in the \(+x\) -direction, and \(\vec{B}\) is \(5.00 \mathrm{~T}\) in the \(-y\) -direction, what are the magnitude and direction of the force that the magnetic field exerts on the charged particle?

Step by step solution

01

Identify given values

From the exercise, we identify charge \(q = -8.00 \times 10^{-6} \, C\), velocity \(\overrightarrow{v} = 3.00 \times 10^{4} \, m/s\) in \(+x\)-direction, and magnetic field \(\overrightarrow{B} = 5.00 \, T\) in \(-y\)-direction.

02

Apply the cross product formula

Applying the formula \(\overrightarrow{F} = q \overrightarrow{v} \times \overrightarrow{B}\), we need to find the cross product of velocity \(v\) and magnetic field \(B\). By the right-hand rule, the direction of \(\overrightarrow{v} \times \overrightarrow{B}\) is \(-z\) (from \(+x\) to \(-y\) gives \(-z\)). Hence, \(\overrightarrow{v} \times \overrightarrow{B} = |v||B|\sin{\Theta}\) represents the magnitude in \(-z\) direction, where \(\Theta\) is the angle between \(v\) and \(B\). Here, \(v\) and \(B\) are perpendicular, so \(\sin{\Theta} = 1\). Substituting, we get \(\overrightarrow{v} \times \overrightarrow{B} = (3.00 \times 10^{4} m/s)(5.00 T)(1) = 1.50 \times 10^{5} T.m/s\) in the \(-z\) direction.

03

Calculate the force

Now we can calculate the force by substituting \(q\) and \(\overrightarrow{v} \times \overrightarrow{B}\) into the formula. This gives force \(\overrightarrow{F} = q \overrightarrow{v} \times \overrightarrow{B} = (-8.00 \times 10^{-6} \, C) (1.50 \times 10^{5} T.m/s) = -1.20 N\) in the \(-z\) direction.

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Key Concepts

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

Moving Charges

A moving charge is simply a particle that carries an electrical charge, such as an electron or proton, that is in motion. When these charges move, they create a magnetic field around them. This concept is fundamental because both electric and magnetic fields originate from electric charges. Whether they are moving or stationary, charges are the source of these fields.

• When a charge is stationary, it creates an electric field around it.
• When a charge starts to move, it not only maintains its electric field but also creates a magnetic field.

In the exercise, the charge is given as \(-8.00 \times 10^{-6} \, C\) moving in the \(+x\)-direction. Understanding the behavior of moving charges in a magnetic field is crucial for solving problems involving magnetic forces.

Cross Product

The cross product is a mathematical operation used to find a vector perpendicular to two other vectors. It is essential in calculating the magnetic force exerted on a moving charge when in the presence of a magnetic field.

• It calculates the vector that represents the force created by the velocity of the particle and the magnetic field.
• Instead of simply multiplying magnitudes, it considers orientation, giving a new vector direction.

In the step-by-step solution, the cross product \(\overrightarrow{v} \times \overrightarrow{B}\) was calculated to be in the \(-z\)-direction. It is this cross product that defines the direction and magnitude of the resulting force vector \(\overrightarrow{F}\). Thus, using the cross product is key to understanding both the direction and strength of magnetic forces on moving charges.

Right-Hand Rule

The right-hand rule is a simple method to determine the direction of the resulting vector in cross-product operations. By applying this rule, you can understand how vectors such as velocity (\(\overrightarrow{v}\)) and the magnetic field (\(\overrightarrow{B}\)) result in a force vector (\(\overrightarrow{F}\)).
To use the right-hand rule:

• Point your fingers in the direction of the first vector (velocity, \(\overrightarrow{v}\)).
• Rotate your wrist, such that your palm can face in the direction of the second vector (magnetic field, \(\overrightarrow{B}\)).
• Your thumb now points in the direction of the resulting cross product vector (\(\overrightarrow{F}\)).

In this specific problem, applying the right-hand rule helps us determine that the force vector points in the \(-z\)-direction. This simple tool is powerful in visualizing and solving problems involving vectors in physics.

Magnetic Field

A magnetic field is a vector field around a magnetic material or a moving electric charge, where force is exerted on other magnetic materials and moving charges. Its presence can thus be detected by the force it exerts on other moving charges.

• The strength of a magnetic field is measured in teslas (T).
• It determines how much force the field can exert on moving charges with velocity \(\overrightarrow{v}\).

In the exercise, a magnetic field of magnitude 5.00 T directed in the \(-y\)-direction interacts with the moving charged particle. Knowing the direction and magnitude of a magnetic field is crucial for predicting the force experienced by a moving charge within it.

Vector Calculation

Vector calculation, especially in physics, involves the careful combination of magnitude and direction to compute various properties like force, velocity, or acceleration.

• Vectors contain both direction and magnitude, which are critical when representing physical quantities like force.
• The sum or product of vectors provides solutions to problems involving paths and forces in space.

In this problem, vector calculations involve using the cross product to determine the direction and magnitude of the magnetic force applied. The correct resolution of the vector forces is vital for correctly interpreting the physical outcome, which in this case, results in a force \(-1.20 \, N\) directed in \(-z\). Mastery of vector operations is indispensable for analyzing and predicting the behavior of objects in a magnetic field.