DP IB Physics: SL

D. Fields

D.3 Motion in electromagnetic fields

DP IB Physics: SL D. Fields D.3 Motion in electromagnetic fields Linking questions:
a)	What causes circular motion of charged particles in a field?
b)	How can the orbital radius of a charged particle moving in a field be used to determine the nature of the particle?
c)	How can conservation of energy be applied to motion in electromagnetic fields?
d)	How are the concepts of energy, forces and fields used to determine the size of an atom?
e)	How are the properties of electric and magnetic fields represented? (NOS)

a) What causes circular motion of charged particles in a field?
Solution:
The magnetic force acting as a centripetal force is what causes charged particles in a field—more especially, a magnetic field—to move in a circular motion.
The magnetic force acting on a charged particle moving perpendicular to a magnetic field is always directed towards the centre of the circular route, which results in the particle moving in a circle.
Figure 1 Motion of charge particle in magnetic field
Magnetic Force:
A force is experienced by a charged particle travelling through a magnetic field.
[math]F = qvBsinθ[/math]
Where q is the charge, v is the velocity, B is the strength of the magnetic field, and is the angle between the velocity and magnetic field vectors, provides the force’s magnitude.
Velocity perpendicular:
[math]F = qvB[/math]
The magnetic force at its highest when the charged particle’s velocity is perpendicular to the magnetic field ([math]θ = 90^o[/math]).
Centripetal Force:
In this instance, the magnetic force functions as a centripetal force, continuously altering the particle’s motion direction (leading to acceleration) without altering its speed. As a result, the particle travels in a circle.
Uniform circular motion:
The particle will move in a uniform circular motion, which means the circular path has a constant radius and the particle moves at a constant speed, if the magnetic field is uniform (constant in strength and direction) and the particle’s velocity is perpendicular to it.
Radius of the Circular route:
The formula [math]r = mv/qB[/math] where m is the particle’s mass, can be used to determine the radius (r) of the circular route.
b) How can the orbital radius of a charged particle moving in a field be used to determine the nature of the particle?
Solution:
The mass-to-charge ratio (m/q) of a charged particle in a magnetic field can be calculated from the radius of its orbital path.
This ratio, which aids in determining the identification of the particle, can be computed by measuring the radius, velocity, and magnetic field strength.
A centripetal force causes a charged particle travelling in a magnetic field to follow a spiral or circular path.
We can identify the particle itself by determining its essential characteristics, such as its mass, charge, or velocity, by measuring the orbital radius of this journey.
Figure 2 Circular motion in a magnetic field
⇒ Magnetic force as Centripetal force:
The magnetic force acting on a moving charge is
[math]F = qvB[/math]
In uniform circular motion, this force provides the centripetal force:
[math]F = \frac{mv^2}{r} \\
qvB = \frac{mv^2}{r} \\
r = \frac{mv}{qB}[/math]
The radius (r) of a charged particle’s circular path in a magnetic field is inversely proportional to the product of its charge (q) and the magnetic field intensity (B), and directly proportional to its momentum (mv). The formula for this relationship is [math]r = mv / qB[/math]
[math]\frac{m}{q} = \frac{Br}{v}[/math]
The mass-to-charge ratio (m/q) can be computed if the particle’s velocity (v) can be ascertained, the magnetic field strength (B) and the orbital radius (r) are known.
The mass-to-charge ratios of various charged particles vary. For instance, in the same magnetic field, a proton and an electron will travel at the same speed but follow different radii.
The type of particle can be determined by comparing the computed m/q with known values for various particles.
c) How can conservation of energy be applied to motion in electromagnetic fields?
Solution:
Although energy can change between many forms, such as kinetic energy, potential energy, and electromagnetic field energy, conservation of energy in electromagnetic fields means that the overall energy within a closed system stays constant.
Fundamentally, the work that electromagnetic forces accomplish on a charged particle is equivalent to the change in the particle’s kinetic energy, and the field gains any energy that the particle loses, preserving the overall energy balance.
According to the rule of conservation of energy, energy can only be changed from one form to another; it cannot be created or destroyed.
This principle still holds true when a charged particle travels through an electric or magnetic field, albeit the type of field affects how energy is transferred.
Figure 3 Conservation of mechanical energy
⇒ Motion in an electric field:
By converting electrical potential energy into motion, an electric field can exert force on a charged particle, altering its kinetic energy.
[math]∆KE = q∆V[/math]
Where:
– q = charge of the particle
– [math]∆V[/math] = Change in electric potential
– [math]\Delta KE = \frac{1}{2}mv^2 [/math] = change in kinetic energy
Energy is conserved:
– Electric potential energy is converted into kinetic energy (or vice versa)
⇒ Energy transformation:
As charged particles pass through an electromagnetic field, their kinetic energy might change. The energy of the electromagnetic field changes in proportion to this energy transfer.
⇒ Closed system:
The overall energy of a system, comprising its kinetic, potential (related to the field), and field energy, stays constant when it is not exposed to outside effects.
d) How are the concepts of energy, forces and fields used to determine the size of an atom?
Solution:
The overall energy of a system, comprising its kinetic, potential (related to the field), and field energy, stays constant when it is not exposed to outside effects.
The equilibrium of repulsive and attractive forces, which are controlled by electrostatic interactions and quantum mechanics, determines an atom’s size.
In particular, an atom’s spatial extension is determined by its electron energy levels, interactions with the nucleus (by electrostatic forces), and effective nuclear charge.
An atom’s size, which is normally around [math]10^{-10}[/math]m, is determined by the balance of forces, fields, and energy rather than by physical barriers like walls. Applying basic physics principles helps us understand atomic size:
⇒ Force and field in the atom:
The electrons (negative) are subject to an attractive electric attraction from the nucleus (positive).
The electric field surrounding the nucleus is the cause of this force:
[math]F = \frac{1}{4\pi\varepsilon_0} \cdot \frac{Ze^2}{r^2}[/math]
– Where Z is the atomic number, e is the elementary charge, and r is the electron’s distance from the nucleus.
⇒ Energy balance determines orbital radius:
The electron’s kinetic energy, which comes from motion, must balance the attracting potential energy in order for it to remain in orbit. As a result, the size becomes stable:
Total energy:
[math]E = KE + PE \\
E = \frac{1}{2}mv^2 – \frac{1}{4\pi\varepsilon_0} \cdot \frac{Ze^2}{r}[/math]
– Only specific quantised energy levels are permitted for electrons in Bohr’s model (for atoms that resemble hydrogen):
[math]r_n = \frac{n^2 h^2 \varepsilon_0}{\pi m e^2 Z}[/math]
– Where n is the energy level (quantum number).
⇒ Quantum Mechanics: Electron cloud instead of orbits:
In contemporary quantum physics:
– Wavefunctions are used to describe electrons.
– The “size” of the atom is determined by the electron’s probability distribution.
– In hydrogen, the most likely distance from the nucleus is represented by the Bohr radius, which is approximately [math]a_0 \approx 5.29 \times 10^{-11} \ \text{m}[/math].
e) How are the properties of electric and magnetic fields represented? (NOS)
Solution:
Field lines are a visual aid used to illustrate the strength and direction of electric and magnetic fields.
The density of electric field lines, which start on positive charges and end on negative charges, indicates the strength of the field.
Conversely, magnetic field lines create continuous loops, and the density of these lines reveals the strength of the field.
The magnetic force acting on a north monopole is indicated by the direction of the magnetic field lines at any given location.
Despite being invisible, vector fields, mathematical formulae, and visual representations can all be used to depict the characteristics of electric and magnetic fields.
According to the Nature of Science (NOS), these representations aid scientists in describing, forecasting, and communicating the behaviour of their domains.
⇒ Electric Fields:
Field Lines:
The electric field surrounding charged objects can be seen using electric field lines.
Direction:
These lines indicate a move towards negative charges and away from positive charges.
Strength:
The electric field’s strength is indicated by the density of field lines. A stronger field is indicated by closer lines.
Representation:
Arrows can also be used to symbolise electric fields; the direction of the arrow indicates the direction of the field, and the length of the arrow indicates the magnitude of the field.
Figure 4 Electric field
⇒ Magnetic Fields:
Field Lines:
Magnetic field lines are used to depict moving charges or the magnetic fields surrounding magnets.
Direction:
A magnet’s magnetic field lines extend from its north pole to its south pole.
Strength:
The magnetic field’s strength is indicated by the density of its lines. A stronger field is indicated by closer lines.
Illustration:
Unlike electric field lines, which end on charges, magnetic field lines are continuous loops.
Right Hand rule:
The direction of the magnetic field surrounding a wire carrying current can be ascertained using the right-hand rule.
Figure 5 Magnetic field with right hand rule