🔄 بنك الأسئلة: التيارات المستحثة

📌 Formula: \(emf=BLv=(0.5)(0.3)(10)=1.5\,V\)

📌 Magnitude: \(B=\dfrac{EMF}{Lv}=\dfrac{0.02}{(0.3)(15)}=4.44\times10^{-3}\,T\)

📌 Direction: the wire moves downward and the current flows A→B as shown; applying \(\vec F=q\vec v\times\vec B\) to positive charges requires the field to point into the page to produce that current direction.

📌 Required emf: \(EMF=IR=(0.5)(10)=5\,V\)

📌 Speed: \(v=\dfrac{EMF}{BL}=\dfrac{5}{(2.5)(0.5)}=4\,m/s\)

📌 Idea: induced current only arises when the magnetic flux \(\Phi_B=B\cdot A\cdot\cos\theta\) through the loop changes. In case B, the loop is entirely within a uniform field and crosses no boundary and does not rotate — its area, orientation, and the surrounding field strength are all constant, so \(\Phi_B\) is perfectly constant. Cases A and C (crossing a field boundary) and D (changing angle via rotation) all change \(\Phi_B\).

📌 Idea: \(V=J/C\), and joule \(J=N\cdot m\) (energy=force×distance), and coulomb \(C=A\cdot s\) (charge=current×time). So: \(V=\dfrac{N\cdot m}{A\cdot s}\)

📌 Idea: a galvanometer works via the "motor effect" (force on a current-carrying coil in a field, \(F=IL\times B\)) — the reverse of induction (we supply current and get motion, not the other way around). Generators and dynamic microphones genuinely use induction (motion produces current), while a mass scale is a purely mechanical device unrelated to electromagnetism at all.

📌 Idea: the vacuum's motor is a rotating coil; as it spins (after the first instant) it also acts as a generator, producing a back-EMF that opposes the original current (Lenz's law), so the net current drops as rotation speeds up, until it settles at the normal running speed.

📌 Formula: \(P_{avg}=\dfrac12 I_m V_m=\dfrac12(2)(12)=12\,W\)

📌 Common mistake: dropping the factor \(\tfrac12\) gives the wrong value 24W — remember the average power of a sinusoid is half the product of the peak values, not their direct product.

📌 From the graph: \(V_m=15\,V\)

📌 Formula: \(P_{avg}=\dfrac{V_m^2}{2R}=\dfrac{15^2}{2(10)}=\dfrac{225}{20}=11.25\,W\)

📌 Formula: \(P_{avg}=\dfrac12 I_m^2 R=\dfrac12(2)^2(6)=\dfrac12(4)(6)=12\,W\)

📌 Average power: \(P_{avg}=I_{rms}V_{rms}=(5)(20)=100\,W\)

📌 Peak power: for a sinusoid, \(P_{max}=2\,P_{avg}=200\,W\) (since instantaneous power oscillates between zero and its peak, averaging to exactly half the peak)

📌 Formula: \(\varepsilon=NBA\omega\sin(\omega t)\) — vanishes at \(\omega t=0°,180°,360°\) (where the coil plane is perpendicular to the field, flux is maximum but its instantaneous rate of change is zero), and is maximum at \(90°,270°\).

📌 Application: stages 1 (0°), 3 (180°), 5 (360°) are the zero-emf stages.

📌 Idea: a generator is driven mechanically (steam, turbine, combustion engine...), rotating a coil within a magnetic field and generating current via induction — i.e., mechanical energy into electrical energy (the exact reverse of an electric motor).

📌 Rule: the induced current (and emf) is maximum when the coil plane is aligned with the field lines (its sides then move perpendicular to the field, cutting the most field lines per unit time).

📌 Application: this recurs every half-turn (180° apart), corresponding to stages A and C in this sequence.

📌 Idea: A is the wave's peak (maximum current, coil parallel to the field), C is the zero-crossing (coil perpendicular to the field), D is the trough (maximum current, opposite direction). Point B lies between peak A and zero C — exactly an intermediate angle on a declining curve, matching the photo's description.

📌 Idea: since both the wire and its velocity are horizontal, the horizontal field component \(B_x\) lies along the wire's line of motion and contributes nothing (\(\vec v\times\vec B_x\) falls outside the wire's own line); only the vertical component \(B_y\) (perpendicular to the entire horizontal plane) actually contributes.

📌 Calculation: \(EMF=B_yLv=(0.3)(0.5)(15)=2.25\,V\)

📌 Lenz's law: the magnet moves away (N pole nearest the coil), so the flux into the coil from that side decreases. The induced current opposes this decrease, i.e. it "tries" to attract the magnet back — so the coil's end facing the magnet becomes a south pole S (since S attracts N).

📌 Idea: if the induced current reinforced the change causing it instead of opposing it, energy would be created from nothing (a runaway self-amplifying feedback with no external source) — violating energy conservation. The "opposition" stated by Lenz's law is thus a necessary condition for energy conservation.

📌 The distinction: Faraday's law gives the magnitude (\(\varepsilon=-d\Phi/dt\)), while the negative sign in it (i.e. the direction) is exactly what Lenz's law specifies.

📌 Lenz's law first: the magnetic force from the induced current must oppose the wire's motion (retard it), i.e. point rightward (opposite the leftward velocity). This alone rules out any figure where the force points the same way as the velocity.

📌 Current direction: applying \(\vec F=q\vec v\times\vec B\) to positive charges (\(\vec v\) left, \(\vec B\) into the page) gives a specific current direction in the wire that matches figure B once combined with the correct force-direction condition.

📌 Idea: a transformer relies on mutual induction, which requires a continuously changing magnetic flux — only possible with an AC source. Figure B is the only one connected to a battery (constant DC), which produces no continuously changing flux (except momentarily at switch-on/off), so it doesn't function as an actual transformer during steady operation.

📌 Idea: wire resistance \(R=\rho L/A\); to minimize power lost as heat (\(P_{loss}=I^2R\)), \(R\) must be minimized — i.e. low resistivity \(\rho\) and large cross-sectional area \(A\).

📌 Idea: stepping up the voltage at the station (and thus stepping down the current, since \(P=IV\) is roughly constant) reduces the \(I^2R\) heat loss along the long transmission lines. Upon reaching the city, voltage is stepped back down (and current up) to a safe level for household use.

📌 Idea: the alternating current in the primary coil produces a changing magnetic flux in the shared core, and this flux in turn induces an emf in the secondary coil — this is exactly the definition of mutual induction.

📌 Idea: when the switch closes, coil 1's current changes from zero to its steady value, changing its own flux — this change induces a back-emf in coil 1 itself (self-induction), while simultaneously changing the flux through the adjacent coil 2 (sharing the same core), inducing an emf there too (mutual induction).

📌 Formula: \(\varepsilon=N\dfrac{|\Delta\Phi|}{\Delta t}=N A\dfrac{|\Delta B|}{\Delta t}=50\times0.02\times\dfrac{0.3-0.1}{0.4}\)

📌 Calculation: \(=50\times0.02\times0.5=0.5\,V\)

📌 Turn area: \(A=\pi r^2=\pi(0.05)^2\approx7.854\times10^{-3}\,m^2\)

📌 Formula: \(\varepsilon=NA\dfrac{\Delta B}{\Delta t}=200\times7.854\times10^{-3}\times\dfrac{0.8}{0.2}\)

📌 Calculation: \(=200\times7.854\times10^{-3}\times4\approx6.28\,V\)

📌 Idea: the formula \(EMF=BLv\sin\theta\) contains no mass term at all — the induced emf is a purely electromagnetic quantity, depending only on geometry and relative motion, not on the wire's inertia.

📌 Formula: \(EMF=BLv\sin\theta=(0.6)(0.4)(5)\sin(30°)=1.2\times0.5=0.6\,V\)

📌 Rule: positive charges feel a force \(\vec F=q\vec v\times\vec B\). Taking \(\hat x\) right, \(\hat y\) up, \(\hat z\) out of the page: \(\vec v=v\hat x\), \(\vec B=-B\hat z\) (into the page).

📌 Calculation: \(\vec v\times\vec B=v\hat x\times(-B\hat z)=-vB(\hat x\times\hat z)=-vB(-\hat y)=vB\hat y\) — the force on positive charges points upward (\(+\hat y\)).

📌 Lenz's law: the approaching N pole increases the flux entering the coil from that side. The induced current opposes this increase, so the face nearest the magnet also becomes a north pole N (to repel it) — and since like poles repel, the force on the magnet is repulsive, resisting its approach (exactly as energy conservation requires: work must be done to push the magnet closer).

📌 Idea: the induced emf is directly proportional to the rate of flux change, which itself is directly proportional to the approach speed (geometry unchanged). So \(\varepsilon\propto v\).

📌 Calculation: \(\varepsilon_2=\varepsilon_1\times\dfrac{3v}{v}=2\times3=6\,V\)

📌 Formula: \(P_{avg}=\dfrac12 I_m^2 R=\dfrac12(6)^2(4)=\dfrac12(36)(4)=72\,W\)

📌 Formula: \(P_{avg}=\dfrac{V_{rms}^2}{R}=\dfrac{100^2}{20}=\dfrac{10000}{20}=500\,W\)

📌 Formula: \(V_s=V_p\left(\dfrac{N_s}{N_p}\right)=220\times\dfrac{100}{1000}=220\times0.1=22\,V\)

📌 Apparent power conservation: \(V_pI_p=V_sI_s \Rightarrow I_s=I_p\left(\dfrac{N_p}{N_s}\right)=0.5\times\dfrac{1000}{100}=0.5\times10=5\,A\)

📌 Check: \(V_pI_p=220\times0.5=110\,W\), and \(V_sI_s=22\times5=110\,W\) ✓ (they match, as required for an ideal transformer)

📌 Idea: \(\varepsilon=-d\Phi_B/dt\) — a constant flux, no matter how large, induces no emf at all (e.g. a stationary loop in a very strong but unchanging uniform field). What matters is the rate of change, not the absolute value.

📌 Idea: in transformers, generators, and motors, eddy currents in the metal cores are an unwanted loss (which is why cores are laminated to reduce them). An induction cooktop, however, works by deliberately generating eddy currents inside the metal cookware itself, which heat it directly via its own electrical resistance — a direct, beneficial use of eddy currents.

📌 EMF: \(EMF=BLv=(0.8)(0.25)(6)=1.2\,V\)

📌 Current: \(I=\dfrac{EMF}{R}=\dfrac{1.2}{2}=0.6\,A\)

📌 Idea: a motor is fed electrical current, producing a force \(F=IL\times B\) on its coil within the field, causing it to spin (electrical→mechanical). A generator, conversely, is driven mechanically, spinning its coil within a field so the flux changes and a current is induced via Faraday's law (mechanical→electrical). Both devices can have nearly identical physical construction (a coil between magnetic poles), but their operating principle is exactly reversed.

📌 (a) Terminals: the current inside the wire (the emf source) flows from A to B, and inside a source current flows from the negative to the positive terminal — so A is negative and B is positive.

📌 Field direction: taking \(\hat x\) from A to B, \(\hat y\) up: velocity \(\vec v=-v\hat y\) (downward). To get a force on positive charges along \(+\hat x\) (toward B), test \(\vec B=-B\hat z\) (into the page, \(\hat z\)=out of page): \(\vec v\times\vec B=(-v\hat y)\times(-B\hat z)=vB(\hat y\times\hat z)=vB\hat x\) ✓ matches the required direction.

📌 (b) General formula: \(EMF=BLv\sin\theta\)

📌 Minimum-field condition: for any given \(EMF\), \(B\) is smallest when \(\sin\theta=1\) (\(\theta=90°\)).

📌 Calculation: \(B_{min}=\dfrac{EMF}{Lv}=\dfrac{0.15}{(0.2)(3)}=0.25\,T\)

📌 (a) With the battery: the transformer does not work. Direct current produces no continuously changing magnetic flux (once it settles), and mutual induction requires a continuously changing flux — so no emf is induced in the secondary during steady operation.

📌 (b) Type of transformer: turns ratio \(N_s/N_p=8/4=2>1\), so it's a step-up transformer.

📌 Secondary voltage: \(V_s=V_p\left(\dfrac{N_s}{N_p}\right)=110\times2=220\,V\)

📌 Secondary current: \(I_s=I_p\left(\dfrac{N_p}{N_s}\right)=2\times\dfrac12=1\,A\)

📌 From the graphs: \(I_m=2\,A\), \(V_m=6\,V\)

📌 (a) rms value: \(V_{rms}=\dfrac{V_m}{\sqrt2}=\dfrac{6}{\sqrt2}\approx4.24\,V\)

📌 (b) Average power: \(P_{avg}=\dfrac12 I_m V_m=\dfrac12(2)(6)=6\,W\). Check: \(P_{avg}=I_{rms}V_{rms}=(I_m/\sqrt2)(V_m/\sqrt2)=\dfrac{I_mV_m}{2}=6\,W\) ✓ (matches)

📌 (a) Parallel motion: No emf is induced at all. Since \(EMF=BLv\sin\theta\) and the angle between the velocity and the field here is \(\theta=0°\) (exactly parallel), \(\sin(0°)=0\), so regardless of how strong the field or speed is, \(EMF=0\).

📌 Moving right (perpendicular to the field): applying \(\vec F=q\vec v\times\vec B\) to positive charges (\(\vec v\) right, \(\vec B\) into the page) induces a current in the wire; the magnetic field produced by this induced current itself (around the wire) opposes the change in the original field caused by the wire's motion — i.e. it resists the increase in flux, per Lenz's law.

📌 Scientist's name: Heinrich Lenz, after whom Lenz's law (for the direction of the induced current/field) is named.

📌 (b) Laminated sheets: the changing magnetic flux within the metal core itself also induces unwanted eddy currents, which dissipate energy as heat (loss). Splitting the core into thin, insulated laminations breaks up the paths available to these eddy currents, confining them to much smaller loops, reducing their magnitude and hence the core's heat loss.