Magnetic flux density

1. Current-carrying wire in a magnetic field:

  • Moving charges cause a magnetic field, which we describe using flux lines.
  • The magnetic flux around a current-carrying wire is shown as concentric circles, indicating the magnitude and direction of the flux pattern.
  • Moving away from the wire, flux lines are further apart because the field gets weaker.
  • If you look at the wire with the conventional current flowing away from you, the flux lines circle the wire in a clockwise direction.
  • Symbols inside the wire indicate the current direction:  indicates a current flowing away from you (Figure 1 a) and . indicates a current flowing towards you (Figure 1 b).

  • Figure 1 The magnetic flux of a current-carrying wire (a) with the current flowing into the page and (b) with the current coming out of the page
  • The combination of flux lines for a loop of wire is shown in Figure 2

  • Figure 2 Magnetic field circles a current carrying wire
  • A current-carrying wire in a magnetic field moves because a force acts on it.
  • The magnetic field making the wire move is called a catapult field.
  • The catapult field is due to the combined effect of the current-carrying wire’s flux and the static flux.

  • ⇒Calculating the direction of current:

  • Magnetic flux density is a vector quantity.
  • When the directions of the magnetic flux, the current in the conductor and the force are all at right angles to each other,
  • Fleming’s left-hand motor rule, shown in Figure 3, helps you see the three-dimensional arrangement of these vectors.
    1. Hold your left hand with your thumb, index finger, and middle finger perpendicular to each other.
    2. Point your thumb in the direction of the current (I).
    3. Your fingers will now be pointing in the direction of the magnetic field (B).
    4. To make it more memorable, imagine that your fingers are “on fire” and the magnetic field is radiating out from your burning fingers.

  • Figure 3 Flaming left’s hand rule

  • ⇒ Calculating the size of the force:

  • The flux density at any point by measuring the force exerted on a current-carrying wire at that point.
  • The tesla is defined as the flux density that causes a force of IN on 1 m of a wire carrying a current of 1 A at right angles to the magnetic field.
  • Written as an equation, this becomes
  • [math] F = Bll [/math]

  • Where F is force (N), B is magnetic flux density (T), I is current (A) and l is the length of the conductor in the field (m)

  • If the wire is at an angle θ to the flux, the force on the wire is calculated using F= Bllsinθ where θ is the angle between the wire, carrying the current, and the flux lines (Figure 4).

  • When the wire lies parallel to the flux lines ( θ = 0) there is no force on the wire.

  • Figure 4 The force on the wire

2. Magnetic flux density:

  • Magnetic flux density (B) is a measure of the strength of a magnetic field. It is defined as the amount of magnetic flux (Φ) per unit area (A) perpendicular to the field.

  • Magnetic Flux Density (B) = Magnetic Flux (∅) / Area (A)

  • The unit of magnetic flux density is the tesla (T), which is defined as:

  • 1 tesla = 1 weber per square meter

  • The tesla is a measure of the magnetic field’s strength and direction. A higher tesla value indicates a stronger magnetic field.
  • Some common examples of magnetic flux density include:
    – Earth’s magnetic field: 0.00003-0.00006 T (30-60 µT)
    – Refrigerator magnet: 0.01-0.1 T (10-100 mT)
    – MRI machine: 1-3 T (1000-3000 mT)
    – Neodymium magnet: 1.3-1.4 T (1300-1400 mT)

3. Force on a charged particle moving in a magnetic field:

  • Charged particles moving in a magnetic field also experience a force.
  • Old-style televisions and computer monitors use electron guns to produce beams of rapidly moving electrons in evacuated tubes, and their direction is controlled using a varying magnetic field.
  • If charge Q travels a distance l in t seconds, then the charge has a velocity[math] v = \frac{l}{t} [/math] . But [math] I = \frac{Q}{t} [/math]  and we can substitute for I in
  • [math] F = Bll [/math]
  • This gives
  • [math] F = \frac{B Q l}{t}\\ F = (BQ)\frac{l}{t} [/math]
  • Since the velocity of the charge is [math] v = \frac{l}{t} [/math], this gives
  • [math] F = BQv [/math]
  • As before, you can use your left hand to predict the direction of the force.
    The thumb represents force, the first finger represents the magnetic field and the second finger represents the direction of a moving positive charge.
    The sign of the charge is important a positively charged particle and a negatively charged particle will move in opposite directions in the same field.
    This is because if a negative charge moves to the left (for example), the conventional current flows to the right.

4. Direction of force on positive and negative charged particles.

  • The direction of the force on positive and negative charged particles in an electric field is determined by the following rules:
    1. Positive charged particles:
      – The force on a positive charge point in the direction of the electric field.
      – The force is in the same direction as the electric field lines.
    2. Negative charged particles:
      – The force on a negative charge point in the opposite direction of the electric field.
      – The force is in the opposite direction to the electric field lines.
  • This can be summarized by the following:
    “Positive particles follow the field, negative particles flee the field”
    Or:
    “Positive particles go with the flow, negative particles go against the flow”
  • This means that if you know the direction of the electric field, you can determine the direction of the force on a charged particle.
  • For example:
    – If the electric field points to the right, a positive charge will experience a force to the right, while a negative charge will experience a force to the left.
    – If the electric field points upwards, a positive charge will experience a force upwards, while a negative charge will experience a force downwards.

5. Cyclotron:

  • A cyclotron is used to force charged particles into a circular path that accelerates them to very high speeds.
  • Cyclotrons are often used with heavier particles like alpha particles and protons.
  • Experiments using particle accelerators investigate the structures of complex molecules like proteins, as well as sub-nuclear structures.
  • The cyclotron is formed from two semi-circular ‘dees’, separated by a small gap and connected to a high- frequency alternating potential difference (Figure 5).
  • A strong magnetic field is applied perpendicular to the dees. The perpendicular magnetic field forces charged particles to move in a circular path inside the dees.

  • Figure 5 Structure of a cyclotron, a proton accelerator.
  • The particles experience a potential difference when they travel across the gap, and gain energy equal to QV (where is the particle’s charge in coulombs, and V is the potential difference in volts).
  • Since the particles have more kinetic energy, they move faster and accelerate to the next dee.
  • The ac voltage is timed to change direction every time the particles reach the gap between the dees.
  • It must alternate to accelerate the particles each time they reach a gap.
  • If the voltage did not alternate, the particles would follow a cycle of accelerate-decelerate-accelerate.
  • Particles spend the same time inside each dee, but the radius of their path increases after each gap and they travel further in the same time.
  • Remember that the centripetal force acting on the charged particle equals the magnetic force on the charged particle, so
  • [math] \frac{mv^2}{r} = BQv [/math]
  • Or
  • [math] \frac{mv}{r} = BQ \\ mv = BQr \\ v = \frac{BQr}{m} \qquad (i) [/math]
  • Because the radius is proportional to the speed of the charged particle., the particles spiral outwards as they accelerate through the cyclotron.
  • The time (t), spent in each dee is given by
  • [math] t = \frac{\pi r}{v} \qquad (ii) [/math]
  • Because πr is half the circumference of the circle. Substituting for v in equation (ii) using equation (i) gives t, the time spent in one dee
  • [math] t = \frac{\pi r}{\frac{B Q r}{m}} \\ t = \frac{\pi m}{B Q} [/math]
  • Which does not depend on either radius or speed.
  • The effect of special relativity limits a particle’s speed in a cyclotron.
  • Particles get more massive as they travel close to the speed of light.
  • As particles move faster and their mass increases, the time spent in each dee increases and the more massive particles get out of step with the alternating potential difference.
  • A synchrotron overcomes this problem by increasing the magnetic field as the speed of the particles increases. The radius of their path remains constant even though the particles travel faster.
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