22.4: Magnetic Field Strength- Force on a Moving Charge in a Magnetic Field (2024)

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    Learning Objectives

    By the end of this section, you will be able to:

    • Describe the effects of magnetic fields on moving charges.
    • Use the right hand rule 1 to determine the velocity of a charge, the direction of the magnetic field, and the direction of the magnetic force on a moving charge.
    • Calculate the magnetic force on a moving charge.

    What is the mechanism by which one magnet exerts a force on another? The answer is related to the fact that all magnetism is caused by current, the flow of charge. Magnetic fields exert forces on moving charges, and so they exert forces on other magnets, all of which have moving charges.

    Right Hand Rule 1

    The magnetic force on a moving charge is one of the most fundamental known. Magnetic force is as important as the electrostatic or Coulomb force. Yet the magnetic force is more complex, in both the number of factors that affects it and in its direction, than the relatively simple Coulomb force. The magnitude of the magnetic force \(F\) on a charge \(q\) moving at a speed \(v\) in a magnetic field of strength \(B\) is given by

    \[F = qvB\sin \theta,\]

    where \(\theta\) is the angle between the directions of \(\bf{v}\) and \(\bf{B}\). This force is often called the Lorentz force. In fact, this is how we define the magnetic field strength \(B\)--in in terms of the force on a charged particle moving in a magnetic field. The SI unit for magnetic field strength \(B\) is called the tesla (T) after the eccentric but brilliant inventor Nikola Tesla (1856–1943). To determine how the tesla relates to other SI units, we solve

    \[B = \frac{F}{qv \sin\theta}\]

    Because \(\sin \theta\) is unitless, the tesla is

    \[1 \,T = \frac{1\, N}{C \cdot m/s} = \dfrac{1\, N}{A \cdot m}\] (note that C/s = A).

    Another smaller unit, called the gauss (G), where \(1 G = 10^{-4} T\), is sometimes used. The strongest permanent magnets have fields near 2 T; superconducting electromagnets may attain 10 T or more. The Earth’s magnetic field on its surface is only about \(5 \times 10^{-5} T\), or 0.5 G.

    The direction of the magnetic force \(\bf{F}\) is perpendicular to the plane formed by \(\bf{v}\) and \(\bf{B}\), as determined by the right hand rule 1 (or RHR-1), which is illustrated in Figure \(\PageIndex{1}\). RHR-1 states that, to determine the direction of the magnetic force on a positive moving charge, you point the thumb of the right hand in the direction of \(v\), the fingers in the direction of \(\bf{B}\), and a perpendicular to the palm points in the direction of \(\bf{F}\). One way to remember this is that there is one velocity, and so the thumb represents it. There are many field lines, and so the fingers represent them. The force is in the direction you would push with your palm. The force on a negative charge is in exactly the opposite direction to that on a positive charge.

    22.4: Magnetic Field Strength- Force on a Moving Charge in a Magnetic Field (2)

    MAKING CONNECTIONS: CHARGES AND MAGNETS

    There is no magnetic force on static charges. However, there is a magnetic force on moving charges. When charges are stationary, their electric fields do not affect magnets. But, when charges move, they produce magnetic fields that exert forces on other magnets. When there is relative motion, a connection between electric and magnetic fields emerges—each affects the other.

    Example \(\PageIndex{1}\): Calculating Magnetic Force: Earth's Magnetic Field on a Charged Glass Rod

    With the exception of compasses, you seldom see or personally experience forces due to the Earth’s small magnetic field. To illustrate this, suppose that in a physics lab you rub a glass rod with silk, placing a 20-nC positive charge on it. Calculate the force on the rod due to the Earth’s magnetic field, if you throw it with a horizontal velocity of 10 m/s due west in a place where the Earth’s field is due north parallel to the ground. (The direction of the force is determined with right hand rule 1 as shown in Figure \(\PageIndex{2}\)).

    22.4: Magnetic Field Strength- Force on a Moving Charge in a Magnetic Field (3)

    Strategy

    We are given the charge, its velocity, and the magnetic field strength and direction. We can thus use the equation \(F = qvB \sin\theta\) to find the force.

    Solution

    The magnetic force is

    \[F = qvB \sin \theta. \nonumber\]

    We see that \(sin \theta = 1\), since the angle between the velocity and the direction of the field is \(90^{\circ}\). Entering the other given quantities yields

    \[ \begin{align*} F &= \left(20 \times 10^{-9} C\right) \left(10 m/s \right) \left(5 \times 10^{-5} T \right) \\[5pt] &= 1 \times 10^{-11} \left(C \cdot m/s \right) \left( \frac{N}{C \cdot m/s} \right) \\[5pt] &= 1 \times 10^{-11} N . \end{align*}\]

    Discussion

    This force is completely negligible on any macroscopic object, consistent with experience. (It is calculated to only one digit, since the Earth’s field varies with location and is given to only one digit.) The Earth’s magnetic field, however, does produce very important effects, particularly on submicroscopic particles. Some of these are explored in the next section.

    Summary

    • Magnetic fields exert a force on a moving charge q, the magnitude of which is \[F = qvB sin \theta , \nonumber \] where \(\theta\) is the angle between the directions of \(v\) and \(B\).
    • The SI unit for magnetic field strength \(B\) is the tesla (T), which is related to other units by \[1 T = \frac{1N}{C \cdot m/s} = \frac{1 N}{A \cdot m}. \nonumber\]
    • The direction of the force on a moving charge is given by right hand rule 1 (RHR-1): Point the thumb of the right hand in the direction of \(v\), the fingers in the direction of \(B\), and a perpendicular to the palm points in the direction of \(F\).
    • The force is perpendicular to the plane formed by \(\mathbf{v}\) and \(\mathbf{B}\). Since the force is zero if \(\mathbf{v}\) is parallel to \(\mathbf{B}\), charged particles often follow magnetic field lines rather than cross them.

    Glossary

    right hand rule 1 (RHR-1)
    the rule to determine the direction of the magnetic force on a positive moving charge: when the thumb of the right hand points in the direction of the charge’s velocity \(v\) and the fingers point in the direction of the magnetic field \(B\) then the force on the charge is perpendicular and away from the palm; the force on a negative charge is perpendicular and into the palm
    Lorentz force
    the force on a charge moving in a magnetic field
    tesla
    T, the SI unit of the magnetic field strength; \(1 T=\frac{1 N}{A⋅m}\)
    magnetic force
    the force on a charge produced by its motion through a magnetic field; the Lorentz force
    gauss
    G, the unit of the magnetic field strength; \(1 G=10^{–4}T\)10–4T
    22.4: Magnetic Field Strength- Force on a Moving Charge in a Magnetic Field (2024)

    FAQs

    22.4: Magnetic Field Strength- Force on a Moving Charge in a Magnetic Field? ›

    F=qvBsinθ, where θ is the angle between the directions of v and B . This force is often called the Lorentz force. In fact, this is how we define the magnetic field strength B —in terms of the force on a charged particle moving in a magnetic field.

    What is the magnetic force on a charge moving in a magnetic field? ›

    The magnetic force is. F = qvB sin θ We see that sin θ = 1, since the angle between the velocity and the direction of the field is 90º.

    What is the expression for force on moving charge in a magnetic field? ›

    The second law of motion by Newton says that the force is equal to the change in momentum per change in the time. For a constant mass, force equals the mass times acceleration, i.e. F = m x a.

    How do you find the force of a charge in a magnetic field? ›

    F=qv×B. This, then, is the Equation that gives the force on a charged particle moving in a magnetic field, and the force is known as the Lorentz force. It will be noted that there is a force on a charged particle in a magnetic field only if the particle is moving, and the force is at right angles to both v and B.

    How do you calculate the strength of a magnetic force? ›

    A current I through a long, straight wire produces a magnetic field with strength H=I/2πr at a distance r from the wire.

    What is the magnetic force exerted by a magnetic field on a charge? ›

    The magnitude of the magnetic force depends on how much charge is in how much motion in each of the objects and how far apart they are. This force is termed as the Lorentz Force. It is the combination of the electric and magnetic force on a point charge due to electromagnetic fields.

    How much force will be experienced by a moving charge in a magnetic field? ›

    The force →F experienced by a particle of charge q moving with a velocity →v in a magnetic field →B is given by →F=q(→v×→B).

    What is the formula for magnetic force? ›

    The magnetic force formula is written as F → = q ∗ v → × B → where q is the charge that is moving with a velocity of v while under the effect of the magnetic field B.

    What is the force on charge in changing magnetic field? ›

    When a charged particle moves along a magnetic field line into a region where the field becomes stronger, the particle experiences a force that reduces the component of velocity parallel to the field. This force slows the motion along the field line and here reverses it, forming a “magnetic mirror.”

    What is the formula for magnetic field strength and force? ›

    We are given the charge, its velocity, and the magnetic field strength and direction. We can thus use the equation F=qvBsinθ to find the force.

    What is the strength of the magnetic field strength? ›

    Represented as H, magnetic field strength is typically measured in amperes per meter (A/m), as defined by the International System of Units (SI). Ampere and meter (or metre) are SI base units constructed from the SI's defining constants. Ampere is the measure of electric current, and meter is the measure of length.

    How do you find the strength of a magnetic field with current? ›

    The strength of a magnetic field, 𝐵 , some distance 𝑑 away from a straight wire carrying a current, 𝐼 , can be found using the equation 𝐵 = 𝜇 𝐼 2 𝜋 𝑑 ,  where 𝜇  is a constant known as “the permeability of free space” and has the value 𝜇 = 4 𝜋 × 1 0 ⋅ /    T m A .

    What is the force on a charged particle in a magnetic field? ›

    Lorentz force, the force exerted on a charged particle q moving with velocity v through an electric field E and magnetic field B. The entire electromagnetic force F on the charged particle is called the Lorentz force (after the Dutch physicist Hendrik A. Lorentz) and is given by F = qE + qv × B.

    What is the force a magnetic field exerts on a charge? ›

    2) The force that a magnetic field exerts on a charged particle is given by F = qo xĒ. A particle with mass m=2.0x10* kg and charge q = +2.5x10-C has an initial speed of v = 4/7 x 102m's moves in a field of 0.5 T.

    What is the force on a charge in magnetic field vector? ›

    The force on a charged particle due to electric charge and magnetic field is given by 'vector F=q vectorE+q vector vvector B. Suppose vector E is along X axis and B along Y axis.

    References

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