Moving charges in a magnetic field (A-level only) — AQA A-Level Physics

This topic explores the magnetic force experienced by a charged particle moving in a magnetic field. You will learn how to calculate the magnitude and direction of this force using the equation F = BQv sin(theta) and Fleming's left-hand rule, and understand why the force is always perpendicular to the velocity, leading to circular motion. The principles are fundamental to many technological applications, from particle accelerators like the Large Hadron Collider to mass spectrometers used in chemistry and medicine.

In the wider context of A-level Physics, this topic builds on your understanding of magnetic fields, electric fields, and circular motion. It bridges electromagnetism and mechanics, and is essential for later topics such as electromagnetic induction and the motion of charged particles in combined electric and magnetic fields. Mastery of this area is crucial for exam success and for pursuing further studies in physics or engineering.

• →The magnetic force on a moving charge is given by F = BQv sin(theta), where theta is the angle between the velocity vector and the magnetic field lines. The force is maximum when the velocity is perpendicular to the field (theta = 90 degrees) and zero when parallel (theta = 0 degrees).
• →The direction of the magnetic force is determined by Fleming's left-hand rule: for a positive charge, the thumb points in the direction of force, the first finger in the direction of the magnetic field, and the second finger in the direction of conventional current (opposite to electron flow). For negative charges, reverse the force direction.
• →The magnetic force does no work on a charged particle because it is always perpendicular to the velocity. Therefore, the speed and kinetic energy of the particle remain constant, and the force acts as a centripetal force, causing uniform circular motion with radius r = mv / (BQ).
• →When a charged particle moves in a magnetic field at an angle other than 90 degrees, its velocity can be resolved into components parallel and perpendicular to the field. The parallel component remains unchanged, while the perpendicular component causes circular motion, resulting in a helical path.
• →The period of circular motion for a charged particle in a magnetic field is T = 2 pi m / (BQ), which is independent of the speed and radius. This is the principle behind the cyclotron, where particles are accelerated in a spiral path by an alternating electric field.

• States that the force acts only on a moving charge and is zero when the velocity is parallel to the magnetic field.
• Applies F = BQv when the field is perpendicular to the velocity, substituting charge, flux density and speed with consistent units.
• Explains that the force is perpendicular to both velocity and field, so it does no work and the speed remains constant.
• Derives or uses BQv = mv²/r to find the radius of the circular path, r = mv/(BQ), for a charged particle in a perpendicular field.
• Determines the direction of the force using Fleming's left-hand rule, reversing the direction for negative charges.
• Describes applications such as the mass spectrometer or cyclotron, linking the radius to mass-to-charge ratio.
• State that the magnetic force on a moving charge has magnitude F = Bqv sin θ, where θ is the angle between velocity and magnetic field, and is zero when the particle moves parallel to the field.
• Apply the left-hand rule correctly: first finger along the magnetic field, second finger along conventional current, thumb giving the force on a positive charge.
• Explain that for a negative charge the conventional current is opposite to the velocity, so the force direction is opposite to that on a positive charge with the same velocity.
• Recognise that the force is perpendicular to velocity, so it acts as a centripetal force, changing direction but not speed or kinetic energy.
• Use a diagram or vector reasoning to determine the force direction for a given field and velocity, including cases where the velocity is at an angle to the field.
• Distinguish between the force on a stationary charge (zero) and on a moving charge, and between motion parallel and perpendicular to the field.
• Derive or use Bqv = mv²/r to show that the radius of the circular path is r = mv/(Bq).
• Explain that the magnetic force is perpendicular to velocity and therefore acts as the centripetal force, so the path is circular at constant speed.
• Show that the period T = 2πm/(Bq) and the cyclotron frequency f = Bq/(2πm) are independent of speed and radius.
• Describe how a cyclotron uses a uniform magnetic field and an alternating potential difference across a gap to accelerate particles in expanding semicircles.
• Explain that the alternating voltage frequency must equal the cyclotron frequency so that particles are accelerated each time they cross the gap.
• Relate the maximum radius of the dee to the maximum kinetic energy or speed of the emerging particles.

• 💡Check the geometry first: if the velocity is parallel to the field, the force is zero, and if perpendicular, use F = BQv directly.
• 💡When asked for the radius, start from the centripetal force condition BQv = mv²/r and rearrange carefully, cancelling v where possible.
• 💡Show the direction reasoning explicitly, stating that the force is perpendicular to both v and B and reversing for negative charge.
• 💡Sketch the field, velocity and force vectors before substituting numbers, so the direction is clear and can be checked against the left-hand rule.
• 💡When the charge is negative, state explicitly that the conventional current is opposite to the velocity before giving the force direction.
• 💡Check whether the velocity is parallel to the field: if it is, the magnetic force is zero and no deflection occurs.
• 💡Start circular-motion questions by equating the magnetic force to the centripetal force, then rearrange for the quantity required.
• 💡When asked about a cyclotron, state the role of the magnetic field (circular motion) and the alternating voltage (acceleration) separately.
• 💡Check units carefully: use kilograms for mass, coulombs for charge and tesla for magnetic flux density before calculating r, T or f.
• 💡Always state the direction of the magnetic force using Fleming's left-hand rule and specify the direction relative to the field and velocity. Marks are often awarded for correct use of the rule and clear indication of direction.
• 💡When deriving the radius of a circular path, show that the magnetic force provides the centripetal force by equating BQv = mv^2 / r. Rearrange carefully and substitute values with correct units. Avoid skipping steps; method marks are available.
• 💡In questions involving combined electric and magnetic fields (e.g., velocity selectors), remember that for a particle to pass undeflected, the electric force must equal the magnetic force: EQ = BQv, leading to v = E/B. This is a common exam scenario.

• Using F = BQv when the velocity is not perpendicular to the field; the correction is to use only the perpendicular component of velocity, or the full vector expression.
• Treating the magnetic force as changing the particle's speed; the correction is to recognise that the force is always perpendicular to velocity, so it changes direction only.
• Applying Fleming's left-hand rule without reversing for a negative charge; the correction is to reverse the predicted force direction for electrons and other negative particles.
• Treating the second finger of the left-hand rule as the direction of motion of a negative particle; the correction is to point the second finger along conventional current, which is opposite to the electron's velocity.
• Assuming the magnetic force can change the speed of a charged particle; the correction is that the force is always perpendicular to velocity, so it does no work and only changes direction.
• Using the right-hand rule for a moving charge; the correction is to use the left-hand rule for the force on a positive charge and then reverse the result for a negative charge.
• Treating F = Bqv sin θ as giving the direction of the force; the correction is that this equation gives only the magnitude, and the direction must be found from the left-hand rule or the vector product.
• Using the magnetic force to change the particle's speed; the correction is that the force is centripetal, so speed and kinetic energy remain constant while direction changes.
• Assuming the cyclotron frequency depends on the particle's speed or radius; the correction is that f = Bq/(2πm) depends only on charge, mass and magnetic flux density.
• Forgetting that the alternating potential difference must reverse each half-cycle; the correction is to match its frequency to the cyclotron frequency so acceleration occurs at every gap crossing.
• Misconception: The magnetic force can change the speed of a charged particle. Correction: The magnetic force is always perpendicular to the velocity, so it does no work and only changes the direction of motion, not the speed or kinetic energy.
• Misconception: The formula F = BQv always applies regardless of the angle between velocity and field. Correction: The correct formula is F = BQv sin(theta). When the velocity is parallel to the field (theta = 0), the force is zero; when perpendicular, it is maximum.
• Misconception: Fleming's left-hand rule applies directly to negative charges without modification. Correction: For negative charges such as electrons, the direction of the force is opposite to that predicted by Fleming's left-hand rule for conventional current. You must reverse the direction of the second finger (or the force) to account for the negative charge.

1. 1Week 1, Day 1-2: Review the basics of magnetic fields and the equation F = BQv sin(theta). Practice determining the direction of force using Fleming's left-hand rule for positive and negative charges.
2. 2Week 1, Day 3-4: Derive and apply the formula for the radius of circular motion, r = mv / (BQ). Solve problems involving protons, electrons, and other charged particles in magnetic fields.
3. 3Week 1, Day 5-6: Study the helical motion of charged particles when velocity is at an angle to the field. Learn about the period of circular motion and its independence of speed.
4. 4Week 2, Day 1-2: Explore applications such as the cyclotron, mass spectrometer, and velocity selector. Work through past paper questions on these applications.
5. 5Week 2, Day 3-4: Complete a full past paper section on this topic under timed conditions. Review mistakes and revisit any misunderstood concepts.

• 📋Calculation of force, radius, speed, or period: You will be given values for B, Q, v, m, etc., and asked to calculate one of these quantities. Always write down the formula, substitute values with units, and give the answer to an appropriate number of significant figures.
• 📋Explanation of circular motion and work done: A common question asks why the magnetic force does no work and how it leads to circular motion. You must state that the force is perpendicular to velocity, so no work is done, and the force acts as a centripetal force.
• 📋Direction of force using Fleming's left-hand rule: You may be asked to determine the direction of the force on a charged particle or to identify the direction of the magnetic field given the force and velocity. Practice applying the rule for both positive and negative charges.
• 📋Combined fields (velocity selector): Questions may involve both electric and magnetic fields, asking for the condition for a charged particle to pass undeflected (v = E/B) or to calculate the speed of the particle.