EMI/AC flash cards

1. 1.Define magnetic flux $\Phi$ through a surface.

   $\Phi = \int \vec{B}\cdot d\vec{A} = BA\cos\theta$, where $\theta$ is the angle between $\vec{B}$ and the area vector (normal to the surface). SI unit: weber (Wb) $=$ T$\cdot$m$^2$.

   Hint: Dot product of field and area vector.

2. 2.State Faraday's law of electromagnetic induction.

   The induced emf equals the negative rate of change of magnetic flux: $\varepsilon = -\dfrac{d\Phi}{dt}$. For $N$ turns, $\varepsilon = -N\dfrac{d\Phi}{dt}$.

   Hint: emf tracks how fast flux changes.

3. 3.State Lenz's law and its physical basis.

   The induced current flows so as to oppose the change in flux that produced it (the source of the minus sign in Faraday's law). It is a consequence of conservation of energy.

   Hint: Nature opposes the change.

4. 4.What are the three ways to change flux and hence induce an emf?

   Change $B$ (field strength), change $A$ (area of the loop), or change $\theta$ (orientation of loop relative to field). Any combination changes $\Phi = BA\cos\theta$.

   Hint: $B$, $A$, or $\theta$.

5. 5.Write the expression for motional emf of a rod of length $l$ moving with velocity $v$ perpendicular to field $B$.

   $\varepsilon = Blv$. More generally $\varepsilon = \int (\vec{v}\times\vec{B})\cdot d\vec{l}$.

   Hint: $Blv$ for mutually perpendicular vectors.

6. 6.A rod moves on rails in field $B$; give the induced current, force needed, and power.

   Current $I = \dfrac{Blv}{R}$; retarding force on rod $F = BIl = \dfrac{B^2l^2v}{R}$; external power to keep constant $v$: $P = Fv = \dfrac{B^2l^2v^2}{R}$, all dissipated as heat.

   Hint: Power in = $I^2R$.

7. 7.What is the origin of motional emf at the microscopic level?

   The magnetic force $q\vec{v}\times\vec{B}$ on free charges in the moving conductor separates them, setting up an electric field until equilibrium; the resulting potential difference is the motional emf.

   Hint: Lorentz force on carriers.

8. 8.What are eddy currents?

   Circulating currents induced in the body of a bulk conductor when the flux through it changes. They dissipate energy as heat and oppose the motion (Lenz's law).

   Hint: Loops of current inside solid metal.

9. 9.Give two applications and one drawback of eddy currents.

   Applications: electromagnetic (induction) braking, induction furnaces, metal detectors, damping in galvanometers. Drawback: energy loss (heating) in transformer/motor cores, reduced by using laminated cores.

   Hint: Braking vs core loss.

10. 10.Why are transformer and motor cores laminated?

    Lamination (thin insulated sheets) breaks up the paths of eddy currents, greatly increasing their resistance and reducing eddy-current heat losses.

    Hint: Thin sheets raise eddy resistance.

11. 11.Define self-inductance $L$.

    The property of a coil by which it opposes any change in its own current: $\Phi = LI$ and $\varepsilon = -L\dfrac{dI}{dt}$. SI unit: henry (H).

    Hint: Flux-linkage per unit current.

12. 12.Give the self-inductance of a long solenoid.

    $L = \mu_0 n^2 A l = \dfrac{\mu_0 N^2 A}{l}$, where $n = N/l$ is turns per unit length, $A$ the cross-sectional area, $l$ the length.

    Hint: Depends on $N^2$ and geometry, not current.

13. 13.Define mutual inductance $M$ between two coils.

    $\Phi_2 = M I_1$ and $\varepsilon_2 = -M\dfrac{dI_1}{dt}$. $M$ depends on geometry, number of turns, and coupling. Unit: henry. Note $M_{12} = M_{21}$.

    Hint: emf in coil 2 from current change in coil 1.

14. 14.Two coils of self-inductance $L_1,L_2$ have mutual inductance $M$. What is the coupling relation?

    $M = k\sqrt{L_1 L_2}$, where the coupling coefficient $0 \le k \le 1$; $k=1$ means perfect (ideal) coupling.

    Hint: Geometric mean of the two self-inductances.

15. 15.Give the energy stored in an inductor carrying current $I$.

    $U = \tfrac{1}{2}LI^2$. This energy is stored in the magnetic field.

    Hint: Analogous to $\tfrac12 CV^2$ for a capacitor.

16. 16.Write the magnetic energy density in a field $B$.

    $u = \dfrac{B^2}{2\mu_0}$ (energy per unit volume). Compare electric: $u_E = \tfrac12\varepsilon_0 E^2$.

    Hint: Field energy per volume.

17. 17.Write the growth of current in an LR circuit switched to a battery of emf $\varepsilon$.

    $I(t) = \dfrac{\varepsilon}{R}\left(1 - e^{-t/\tau}\right)$, with time constant $\tau = L/R$.

    Hint: Rising exponential to $\varepsilon/R$.

18. 18.Write the decay of current in an LR circuit when the battery is shorted out.

    $I(t) = I_0\, e^{-t/\tau}$, with $\tau = L/R$ and $I_0$ the initial current.

    Hint: Decaying exponential.

19. 19.What is the time constant of an LR circuit, and its physical meaning?

    $\tau = L/R$ (seconds). It is the time for the current to reach $\approx 63\%$ of its final value during growth (or fall to $37\%$ during decay).

    Hint: $L$ over $R$.

20. 20.During current growth in an LR circuit, at $t=\tau$ what fraction of the final current is reached?

    $1 - e^{-1} \approx 0.632$, i.e. about $63.2\%$ of the steady value $\varepsilon/R$.

    Hint: $1 - 1/e$.

21. 21.Why can't current in an inductor change instantaneously?

    An instantaneous change would require infinite $dI/dt$, hence infinite back-emf $L\,dI/dt$. So inductor current is continuous; it acts as a short at long times (DC) and opposes sudden changes.

    Hint: Back-emf forbids jumps in $I$.

22. 22.Write the equation of a sinusoidal alternating emf and current.

    $\varepsilon = \varepsilon_0\sin\omega t$, $I = I_0\sin(\omega t + \phi)$, where $\omega = 2\pi f$ is angular frequency and $\phi$ the phase difference.

    Hint: $\omega = 2\pi f$.

23. 23.Define the rms value of an alternating current.

    The rms (root-mean-square) value is the steady DC current that produces the same average heating: $I_{rms} = \sqrt{\langle I^2\rangle}$. For sinusoid $I_{rms} = I_0/\sqrt{2}$.

    Hint: Equivalent DC for heating.

Open the interactive deck for the other 66 cards, with self-grading so the ones you keep missing come back.