Motion of Charged Particles in a Magnetic Field

Transcription

1 Motion of Charged Particles in a Magnetic Field

2 Last Chapter: In the previous chapter, we considered the force on a current carrying wire in a magnetic field. F B = IlB sin θ Electric current I is the measure of the rate of flow of electric charge: I avg = Δq Δt

3 On the microscopic scale, electric current is realised as the motion of charged particles. Remember, the direction of conventional current is defined as the direction positive charges flow. So for the diagram above, the current is to the? Right

4 In principle, a single moving charge constitutes a current. Let us consider an alpha particle passing through a point P At time t 1, the alpha particle is at A and has velocity v P + + A

5 In principle, a single moving charge constitutes a current. Let us consider an alpha particle passing through a point P At time t 1 = 1.0 s, the alpha particle is at A and has velocity v At some later time t 2 = 4.0 s, the alpha particle is now at B A P + + B In the time interval, t = t 2 t 1 = 3.0 s, the alpha particle passed through the point P. So the average current during this time interval is: I avg = q t = C 3.0 s = A

6 Force on a moving charge? If a moving charge can constitute a current, then we should expect that a moving charge must experience a force when moving through a magnetic field. This is indeed the case! Moving charges in a magnetic field experience a force. In fact we can derive (not examinable) the equation for this force from what we already know.

7 We know that the force on a current carrying wire is given by: F B = IlB sin θ This force can be understood as the cumulative force acting upon all the moving charged particles that constitute the current I. If N particles of charge q pass a given point in a time interval t, then the current is I = Nq t How do we define the length of the current element l?

8 Consider observing the motion of one of these charged particles in the time interval t During this time interval, the particle will travel a distance t 2 l = v t t 1 So: F B = IlB sin θ v θ = Nq t v t B sin θ q = N qvb sin θ

9 Lorentz Force Law The force that acts upon a charged particle is described by the Lorentz force law: F B = qvb sin θ Where: q is the particles electric charge v is the particles velocity B is the magnetic field strength θ is the angle between v and B

10 The angle θ is the angle between the particles velocity v and the magnetic field B q θ v B As was the case with the force on a current carrying wire: the maximum force occurs when θ = 90 ( v B) and the minimum when when θ = 0 ( v B) In particular, the minimum force is? ZERO!

11 Graph of sin θ vs. θ sin θ θ

12 No magnetic force So we have noted that when v B the resulting magnetic force is zero as sin θ = 0 q v v B In what other situation will the resulting force be zero? Hint: One word we ve seen quite frequently is moving When the particle is stationary, the resulting force is zero as v = 0.

13 Some worked examples 1. A proton travelling with a speed v = m s 1 enters a uniform magnetic field of strength T. Calculate the force on the proton if: i. The proton s direction of travel is perpendicular to the magnetic field ii. The proton s direction of travel is anti-parallel to the magnetic field iii. The proton s direction of travel is at an angle of 56 to the direction of the magnetic field 2. What would be the difference in force in Q1. iii. if the proton were instead an electron? 3. What would be the difference in the acceleration?

14 Some more worked examples 4. A positron experiences a force of N as it passes through a magnetic field. Given that it s speed is m s 1 and it s direction of travel is at an angle of 70, what is the magnitude of the magnetic field? 5. A velocity selector operates by having a opposing electric and magnetic forces as the charge travels the device. Determine an alpha particles speed if the electric field strength in the velocity selector is 0.1 V m 1 and the magnetic field strength is 0.1 mt. Note that the alpha particle is moving perpendicularly to the magnetic field.