PHYSICS 1B Today s lecture: Motion in a Uniform Magnetic Field continued Force on a Current Carrying Conductor Introduction to the Biot-Savart Law Electricity & Magnetism
A Charged Particle in a Magnetic Field Consider a particle moving in an external magnetic field with its velocity perpendicular to the field. The force the particle experiences is always directed towards the centre of the circular path. In other words: The magnetic force causes a centripetal acceleration, which changes the velocity of the particle. Remember: the speed of the particle doesn t change.
Force on a Charged Particle Use the particle under a net force and particle in uniform circular motion models (from mechanics) Equating the magnetic and centripetal forces: Solve for r: F B = qvb = mv2 r r = mv qb r is proportional to the particle s linear momentum r is inversely proportional to the magnetic field
A Charged Particle s Motion We can go further than this. The angular speed of the particle is clearly: ω = v r = qb m ω is often called the cyclotron frequency. The period of the motion is therefore: T = 2πr v = 2π ω = 2πm qb
A Charged Particle s Motion: General If our charged particle is moving in a magnetic field at some arbitrary angle with respect to the field, then the path it follows will be a helix. For the case where the field is in the +x direction: The same equations as before apply, but with the velocity v replaced by v v v 2 2 y z This gives the motion of the particle around the centre of the helix its velocity in the x-direction does not change.
A Charged Particle s Motion: Examples The bending of an Electron Beam Electrons are accelerated from rest through a potential difference. These electrons then enter a uniform magnetic field that is perpendicular to their velocity vector. The electrons then travel in a curved path. We can use conservation of energy to find v, and from this we can the work out other parameters. See example 29.3 in the text book if you want to try working this one out!
A Charged Particle s Motion: Examples The Van Allen Radiation Belts The Van Allen radiation belts consist of charged particles surrounding the Earth in doughnut-shaped regions. Existence confirmed by James Van Allen, who was working on the Explorer 1 and 3 missions. The particles are trapped by the Earth s nonuniform magnetic field. The particles spiral from pole to pole, and can create aurorae when they hit the Earth s atmosphere.
The Earth s Magnetosphere and the Van Allen belts
Charged Particles Moving in E and B Fields. In many applications, charged particles will move in the presence of both magnetic and electric fields. In such cases, the total force experienced by the particle is the sum of the forces due to the individual fields. The total force experienced by the particle due to the magnetic and electric fields is known as the Lorentz force. In general: F = qe + qv B
The Velocity Selector A velocity selector is a device used in experiments which require charged particles that are all moving at the same velocity. A uniform electric field is perpendicular to a uniform magnetic field. When the force due to the electric field is equal in magnitude, but opposite in direction to the force due to the magnetic field, the particles will move in a straight line.
The Velocity Selector Particles will move in a straight line when: v = E B Only the particles with the chosen speed will pass through undeflected. The fastest particles will deflect to the left (since for them, F B > F E Slow moving particles will deflect to the right.
The Mass Spectrometer A mass spectrometer separates ions according to their mass-to-charge ratio. In the design to the right, a beam of ions passes through a velocity selector (so all have the same velocity). Those ions then enter a second magnetic field. In that field, the ions move in a semicircle of radius r before they strike a detector at point P.
The Mass Spectrometer Positively charged ions deflect to the left. Negatively charged ions deflect to the right. Check this with the right hand rule! The mass to charge ratio (m/q) can be determined if we know E and B, just by measuring r. m q = rb 0 v = rb 0B E
Measuring the charge-mass ratio of the electron J. J. Thomson measured the charge-mass ratio of the electron in 1897, using the apparatus to the right. Electrons are accelerated from the cathode. They are then deflected by electric and magnetic fields. The beam of electrons strikes a fluorescent screen. By measuring the beam s deflection, it was possible to calculate e/m!
Quick Quiz A charged particle is moving perpendicular to a magnetic field in a circle, with radius r. An identical particle enters the field, again with v perpendicular to B, but with a greater speed, v, than the first particle. Compared to the radius of the circle followed by the first particle, the radius of the circle for the second particle is: a) Smaller b) Larger c) Equal in size
Quick Answer A charged particle is moving perpendicular to a magnetic field in a circle, with radius r. An identical particle enters the field, again with v perpendicular to B, but with a greater speed, v, than the first particle. Compared to the radius of the circle followed by the first particle, the radius of the circle for the second particle is: b) Larger The magnetic force on the particle increases in proportion to v, but the centripetal acceleration increases according to the square of v. Taken together, this results in a larger radius, as we can see from: r = mv qb
Quick Quiz A charged particle is moving perpendicular to a magnetic field in a circle, with radius r. The magnitude of the magnetic field is increased. Compared to the initial radius of the circular path, the radius of the new path is: a) Smaller b) Larger c) Equal in size
Quick Answer A charged particle is moving perpendicular to a magnetic field in a circle, with radius r. The magnitude of the magnetic field is increased. Compared to the initial radius of the circular path, the radius of the new path is: a) Smaller The magnetic force on the particle increases in proportion to B. This results in a smaller radius for the circle followed by the particle. Again: r = mv qb
Quick Quiz Three types of particle enter a mass spectrometer like that to the right. The figure below shows where the particles hit the detector. Rank the particles that arrive at a, b and c by speed. a) a, b, c b) b, c, a c) c, b, a d) All the speeds are equal
Quick Answer Rank the particles that arrive at a, b and c by speed. d) All the speeds are equal The velocity selector ensures that all three types of particle have the same speed.
Quick Quiz Now, rank the particles that arrive at a, b and c by their mass to charge ratio, m/q, from largest to smallest. a) a, b, c b) b, c, a c) c, b, a d) All the m/q ratios are equal
Quick Answer Now, rank the particles that arrive at a, b and c by their mass to charge ratio, m/q. c) c, b, a We can not determine individual masses or charges, but we can rank the particles in m/q. The particles that go through the largest circle must have the largest m/q ratio
Electric Current Most practical applications of electricity deal with electric currents flows of charge from one region of space to another. Electric current is the rate of flow of charge through some region of space. The SI unit of current is the ampere (A). 1 A = 1 C / s The symbol for electric current is I.
Force on a Current Carrying Conductor A force is exerted on a current-carrying wire placed in a magnetic field. The current is a collection of many charged particles in motion. The direction of the force is given by the right hand rule
Force on a Wire In the figure to the right, no current is flowing through the wire. Since no charge is moving within the wire, no force is exerted on it by the magnetic field. The wire therefore remains vertical.
Force on a Wire In the figure to the right, a current is flowing through the wire from the bottom to the top. By definition, current flows from positive to negative (even though normally it is carried by electrons, which flow from negative to positive!). As such, the direction current flows in is the direction that a positive charge would move. Therefore, the right hand rule shows us that the wire deflects to the left.
Force on a Wire In the figure to the right, current is now flowing through the wire from the top to the bottom. The force is therefore to the right (right hand rule, again). The wire therefore deflects to the right.
Force on a Wire: The Maths Magnetic force is exerted on each moving charge inside the wire. That force is given by: F = qv d B The total force is the product of the force on one charge and the number of charges, i.e.: F = qv d B nal where n is the number of charges per unit volume, A is the cross sectional area of the wire, and L the length of the segment.
Force on a Wire: The Maths F = qv d B nal Now the current, I, is given by the number of charges (of charge q) flowing through area A with velocity v D. We can therefore simplify the equation by substituting in that I = nqav D To give us: F = IL B where L is a vector that points in the direction of the current, and whose magnitude is the length, L of the segment.
Force on a Wire of Arbitrary Shape Consider a small segment of the wire, ds. The force exerted on that segment is: df B = I ds B To work out the total force, we just integrate between the points of interest, i.e.: F B = I b ds B a
Force on a Wire: Case 1 Suppose that the field, B, is uniform. Our integral Becomes b F B = I ds B a F B = IL B (since the integral from a to b of ds is the vector sum of all length elements from a to b i.e. the line between a and b). The magnetic force on a curved current carrying wire in a uniform field is equal to that on a straight wire connecting the end points and carrying the same current.
Force on a Wire: Case 2 Again, consider a uniform field, B. Our integral b F B = I ds B a This becomes F B = I ds B = 0 The length elements here form a closed loop, so their vector sum is zero. Therefore, the net magnetic force acting on any closed current loop in a uniform magnetic field is zero.
Quick Quiz The four wires shown to the right all carry the same current from point A to point B through the same magnetic field. Which of the following choices ranks them according to the magnitude of the magnetic force exerted on them, from the greatest force, to the least? a) a, b, c b) a, c, b c) a, c, d d) a, d, c e) b, c, d f) c, b, d
Quick Answer Which choices ranks them according to the magnitude of the magnetic force exerted on them, from the greatest force, to the least? c) a, c, d The order is (a), (b) = (c), (d). The magnitude of the force depends on the value of sin θ. The maximum force occurs when the wire is perpendicular to the field i.e. case (a). There is zero force in case (d), when the wire is parallel to the field. Choices (b) and (c) represent the same force because a straight wire between A and B will have the same force on it as the curved wire, for a uniform field.