Force on Conductor & Galvanometer Force on Conductor & Galvanometer A magnetic field does not just sit quietly around a current; it acts on it. When charges move through a wire placed in a magnetic field, each charge feels a sideways magnetic push. All these tiny pushes add up to a definite force on the wire. If the wire forms a loop, the forces on opposite sides are equal and opposite, so the net force can be zero even though a twist (torque) tries to rotate the loop. This simple push-and-twist is the working heart of meters you use in circuits: the moving-coil galvanometer. A light coil carrying current sits in a magnetic field and experiences a torque. A spring resists this twist. The balance between magnetic torque and spring torque gives a deflection that is proportional to the current. Once you understand force on a straight conductor and torque on a loop, it becomes straightforward to design an ammeter by putting a low shunt in parallel with the coil or a voltmeter by adding a high series resistor. The ideas are simple, but details like which angle goes into sinθ, what counts as the area vector, and which current is Ig (full-scale current) cause many NEET traps. This lesson builds carefully from intuition to formulas, then applies them in clean numerical steps so you can avoid those traps and compute answers confidently. Analogy: Think of a paddle wheel in flowing water. The water’s push does not drag the wheel linearly but twists it. Similarly, a magnetic field does not “pull” charges along the wire; it pushes them sideways, producing a net torque on a loop. remember The net magnetic force acting on a current-carrying wire placed in a magnetic field. For a straight segment of length L in uniform B, magnitude is F = B I L sinθ. Magnetic force on a conductor Angle θ The angle between the direction of the conductor’s length vector (in the direction of current) and the magnetic field vector B. A couple acting on a current loop in a magnetic field that tends to rotate it. Magnitude τ = n B I A sinθ, where θ is angle between area vector and B. Torque on a current loop For a current loop, μ = n I A with direction given by the right-hand rule (curl fingers with current; thumb gives μ). Magnetic dipole moment (μ) Moving-coil galvanometer A sensitive device that detects and measures small currents by the deflection of a current-carrying coil in a magnetic field. The current needed per unit deflection for a galvanometer: k = I/θ. Smaller k means higher current sensitivity. Figure of merit (k) A small resistance connected in parallel with the galvanometer coil to convert it into an ammeter of higher range. Shunt ( R s ) A high resistance placed in series with the galvanometer to convert it into a voltmeter of higher range. Multiplier/Series resistor (R series) Start with the microscopic Lorentz force on a single moving charge: a charge q moving with velocity v in a magnetic field B experiences a force q( v × B ). Inside a wire, many charges move together, giving a current I. If the magnetic field is uniform and the conductor is a straight segment of length L, the vector sum of forces on all charges reduces cleanly to a single force on the segment. The direction follows a right-hand rule: point your index finger along the current, your middle finger along B, and your thumb then points along the force on a positive charge. The magnitude depends on the sine of the angle between the wire’s length direction and the magnetic field lines, reaching maximum when they are perpendicular and vanishing when they are parallel. Magnitude F = B I L ; direction by right-hand rule for cross product. Force on a straight conductor In the magnitude form F = B I L sinθ, the length L is the straight-line vector from one terminal of the segment to the other, along the current direction. If the wire curves, split it into elements dl and integrate. Edge cases: if θ = 0° or 180°, sinθ = 0 and there is no force; if θ = 90°, sinθ = 1 and the force is maximum. Also notice that the force does not do work on charges moving perpendicular to it, which is consistent with magnetic forces changing direction of motion but not speed in ideal cases. Applicability: F = B I L sinθ requires a straight segment in uniform B. For arbitrary shapes, use d F = I d l × B and integrate over the conductor. tip Elemental force Integrate over the entire wire to get net force for arbitrary shapes. Use this law to calculate the magnetic field produced at a point by any segment of a current-carrying conductor. For a closed loop in a uniform magnetic field, the net force is zero. This is because the integral of d l over a closed path is zero, and more precisely, the vector sum of I d l × B around the loop cancels. Yet the loop still experiences a torque: forces on opposite sides form a couple. That torque tends to rotate the loop so that its area vector aligns with the magnetic field. = n B I A Uniform magnetic field B. Rigid rectangular coil of area A with n turns. Angle ( ) is between area vector and ( B ). Quasi-static balance (no large acceleration). Here ( ) is the angle between side and ( B ); this relates to ( ) between area vector and ( B ). Area A = a b. Hence = B I A for one turn. Direction by = B , with = n I A . Torque on a rectangular coil: ( = n B I A ) The torque on a loop is compactly written using the magnetic dipole moment: define μ = n I A with direction normal to the loop given by the right-hand rule. Then τ = μ × B and potential energy U = − μ · B. The loop seeks to reduce its potential energy, so it rotates to align μ with B. Equilibrium positions: stable when μ is parallel to B (θ = 0), unstable when μ is antiparallel to B (θ = π). Maximum torque occurs at θ = 90°. Compact vector form for torque and energy of a current loop. Dipole form This energy determines the stability of the loop, showing minimum potential energy when the magnetic dipole aligns parallel to the external field. These relations are universal for planar current loops in uniform magnetic fields. For non-uniform fields, a net force on the loop can appear, often approximated as ( F ( ) B ) in advanced treatments. In this course, focus on uniform fields: net force zero on a closed loop, but a torque exists unless θ = 0 or π. Classic trap: In F = B I L sinθ, θ is the angle between L (in current direction) and B, not the angle between the wire’s plane and B. In τ = n B I A sinθ, θ is between area vector and B, not between the wire and B. neet-alert A moving-coil galvanometer turns magnetic torque into a readable deflection. The coil of N (or n) turns and area A is placed in a strong, nearly radial magnetic field produced by a curved pole piece and a soft iron core. Radial field keeps B perpendicular to the coil’s plane over its motion so that torque is nearly proportional to current for a wide range of angles. The coil is suspended by a thin phosphor-bronze strip or mounted on jeweled bearings. A hair spring provides restoring torque. When current flows, magnetic torque n B I A balances spring torque k θ at equilibrium; the pointer attached to the coil shows an angle proportional to current. Deflection law Linear scale for small angles: ( I ). This law applies when the magnetic torque acting on a current-carrying coil is balanced by the restoring torque provided by the suspension s Coil resistance (G) The internal resistance of the galvanometer coil and its leads. Often denoted G and used in range-conversion formulas. Deflection per unit current: S i = θ/I = n B A / k. Larger S i means more deflection for the same current. Current sensitivity ( S i ) Voltage sensitivity ( S v ) Deflection per unit voltage: S v = θ/V = (n B A)/(k G). Increasing coil resistance G decreases S v . From the deflection law n B I A = k θ, current sensitivity is S i = θ/I = n B A / k. If V is applied across the coil, I = V/G so θ = (n B A / k)(V/G), giving voltage sensitivity S v = θ/V = (n B A)/(k G). To increase S i , we can increase n, B, and A, or decrease k. However, increasing n raises G and coil inertia; increasing A enlarges the instrument; lowering k may reduce stability. Voltage sensitivity decreases when G increases, so there is a trade-off between sensitivity and power dissipation. tip Design tip: Radial magnetic fields (curved pole faces) make the scale linear over larger angles because the field remains nearly perpendicular to the coil for all θ. To extend the range for measuring large currents, convert the galvanometer into an ammeter. The idea is to bypass most of the current through a low resistance shunt R s connected in parallel with the coil. At full-scale reading, the galvanometer can safely carry only I g (the full-scale deflection current). If the desired ammeter range is I (≫ I g ), then the remainder I − I g must flow through the shunt. The parallel connection ensures both branches have the same voltage drop, allowing a simple, exact expression for R s . R s = G , I g I - I g Kirchhoff’s current law at the junction. Parallel branches have equal potential difference. Shunt is small because I s is large compared to I g . Shunt for ammeter: (R s = G , I g I - I g ) Ideal connections; temperature effects ignored. Galvanometer coil resistance is G. Full-scale deflection at I g through the coil. Desired ammeter full-scale is I. To convert a galvanometer into a voltmeter, place a large series resistance R series so that only a small current I g flows at the full-scale voltage V. Because current through series elements is the same, the deflection law still applies. The series resistor drops most of the applied voltage. The resulting formula directly tells you what resistance to add to reach any desired voltage range, keeping the coil safe. R series = V I g - G Series resistor for voltmeter: (R series = V I g - G ) Single-loop series circuit. Uniform current through series elements. Large R series ensures small coil current. Ideal meter with coil resistance G. Full-scale deflection at current I g. Desired full-scale voltage is V. High-resistance voltmeters draw little current, minimizing loading of the circuit under test. Ammeters, on the other hand, must have very low resistance to avoid altering the measured current. In practical instruments, damping (air vane or eddy currents) is added to make the pointer settle quickly without oscillation. Temperature changes can alter G; careful design uses materials with low temperature coefficients. Ammeter In series with load Very low Low shunt in parallel with coil R s = (G I g )/(I - I g ) Most current bypasses the delicate coil Voltmeter In parallel across load Very high High series resistor R series = V/I g - G Draws tiny current; minimal loading Instrument Connection Desired total R Added resistor Key formula Why Identify coil parameters: G and I g (full-scale current). Choose target range: I for ammeter, or V for voltmeter. Use the correct formula to compute R s or R series. Check power ratings: P = I 2 R for shunt/series parts. Verify sensitivity and scale linearity in the chosen field. Design steps overview Modeled on standard NCERT exercise style A straight wire of length 5.0 cm carries a current of 3.0 A. It lies in a uniform magnetic field of 0.20 T making an angle of 30° with the wire. Find the magnitude and direction of the magnetic force on the wire. Use F = B I L ; direction by right-hand rule (current × B). Magnitude F and direction easy L = 5.0 10 -2 , m I = 3.0 , A B = 0.20 , T = 30 The formula F = B I L sinθ treats the wire as a single vector segment. The direction uses the cross product: if current points along your index finger and B along your middle finger, your thumb shows the force on a positive charge and therefore the direction of force on the conductor. If the current direction is reversed, the force reverses. n = 50 a = 2.0 10 -2 , m , b = 4.0 10 -2 , m A = a b = 8.0 10 -4 , m 2 I = 0.20 , A B = 0.50 , T Plane makes 30 with B = 60 medium Torque magnitude τ and whether net force is zero Use = n B I A . For a uniform B, net force on a closed loop is zero. A rectangular coil of 50 turns has dimensions 2.0 cm × 4.0 cm and carries 0.20 A. It is placed in a uniform 0.50 T magnetic field so that its plane makes an angle of 30° with the field. Find the torque on the coil and state whether the net force on the coil is zero or not. Standard torque-on-coil computation The angle trap appears again: if the plane makes 30° with B, the area vector makes 60° with B, hence sinθ = sin60°. In all such problems, first convert the given orientation into the angle between the area vector and B. Also note that for a uniform magnetic field, the net force on a closed loop is zero, though the torque is generally non-zero. Composite NEET-style conversion problem Use R s = (G I g )/(I - I g ) and R series = V/ I g - G. Also S i = n B A/k and S v = (n B A)/(k G). A moving-coil galvanometer has coil resistance G = 50 Ω and requires I g = 100 μA for full-scale deflection. (a) What shunt R s is needed to make a 10 A ammeter? (b) What series resistance R series is needed to make a 10 V voltmeter? Also find the current and voltage sensitivities of the bare galvanometer if n B A/k = 2.0 × 10 4 rad/A. hard (a) R s , (b) R series, (c) S i and S v of the bare coil G = 50 , I g = 100 10 -6 , A I = 10 , A ( for ammeter ) V = 10 , V ( for voltmeter ) n B A/k = 2.0 10 4 , rad/A Be careful with full-scale current I g . It is the maximum safe current through the coil at full deflection, not the desired ammeter range I. When designing the shunt, almost the entire current flows through R s . Also note that in the voltmeter design, the series resistor dominates the total resistance; rounding to 1.0 × 10 5 Ω is acceptable to two significant figures. custom Angular dependence of torque on a planar current loop in uniform B. Torque τ (arb. units) control dependent τ = sin θ Stable equilibrium (τ = 0) Maximum torque π/2 Unstable equilibrium (τ = 0) Torque on a loop varies as sin θ, peaking at θ = 90° and zero at 0 and π. Angle θ (rad) 2D PLOT Torque on a current loop vs orientation angle tau = tau max sin(theta) theta tau tau max Max torque If the plane is parallel to B, the area vector is perpendicular to B (θ = 90°), so torque is maximum, not zero. If the plane of the loop is parallel to B, torque is zero. In ammeter conversion, the galvanometer current equals the ammeter range I. The galvanometer only carries I g at full-scale; the rest (I − I g ) is diverted through the shunt. Using I instead of I g in the formula gives a wrong shunt value. Units trap: I g is often in microampere. Convert to ampere before using R s = (G I g )/(I − I g ). Similarly, in torque or force problems, convert cm to m, mT to T, and degrees to radians when needed. neet-alert Series for Voltmeter, Shunt for Ammeter: SV–SA. Pair it with properties: Voltmeter → Very high R (Series); Ammeter → Almost zero R (Shunt). SV–SA quick recall Direction of force on a conductor (right-hand rule) Point index finger along the current I. Point middle finger along the magnetic field B. Your thumb now gives the direction of the force on the conductor (for positive charges). The force between two long parallel current-carrying wires is an application of the same idea: each wire creates a magnetic field that exerts a force on the other. If currents flow in the same direction, the wires attract; if opposite, they repel. While the detailed formula and the definition of the ampere are covered elsewhere, remember the physical picture: a conductor in the magnetic field of another conductor experiences a sideways force. This formula describes the net force acting on a magnetic dipole moment ( ) when it is placed in a spatially varying (non-uniform) If the magnetic field is non-uniform, a loop can experience a net force in addition to torque. The side of the loop in the stronger field feels a larger force, shifting the loop’s center of mass. This is the basis of how small magnetic dipoles move in inhomogeneous fields (qualitatively: they drift toward stronger fields if aligned appropriately). In NEET-level problems, unless stated, fields are usually treated as uniform over the loop’s extent. Qualitative non-uniform field effect Beyond scope for detailed computation here; included to frame intuition. Damping in a galvanometer ensures quick settling. Air damping uses a lightweight vane moving in air; eddy-current damping uses a conductor moving in a magnetic field to create currents that oppose motion. Critical damping avoids overshoot but reaches equilibrium quickly. Overdamping slows too much; underdamping causes oscillations. Designers pick materials and vane sizes to balance speed and smoothness. Common error sources include temperature dependence of the coil resistance G, friction in pivots (if used), stray magnetic fields, and contact resistance at terminals. Zero the pointer before use and avoid strong external fields. For precision, shielding with soft iron and using low-temperature-coefficient wire helps maintain calibration. Worked examples suggest a general solution pattern: translate geometry into the correct angle, write the appropriate formula (force for straight segments, torque for loops), insert values with consistent SI units, and interpret the direction using the right-hand rule. In conversion problems, start from the coil’s safe current I g and resistance G, then size the shunt or series resistor using the core formulas. Finally, check that the resulting resistance is physically sensible (very small for shunt, very large for series). Advanced note on scale linearity: In a uniform, non-radial field, τ = n B I A sinθ would give a sin dependence with θ, causing non-linear deflection. The moving-coil design with a radial field makes the effective torque nearly independent of θ because B is designed to stay perpendicular to the coil sides, resulting in θ ∝ I for a wide range of angles and a convenient linear scale. Big-picture: Magnetic force changes direction of motion (or produces torque) but does not do work in ideal conditions. The energy change you see on the pointer comes from electrical power supplied to the coil and dissipated as heat; the magnetic field guides the motion. remember Verify resistor ratings in range conversion. Power checks This formula set applies to analyzing the power dissipated in the three components (coil, shunt, and series resistor) of a galvanometer when Before finalizing a design, verify thermal safety. The shunt must handle nearly the full current without overheating. The series resistor in a voltmeter can have a large voltage drop and hence significant power dissipation even at small currents; use P = I g 2 R series to choose rated components. Thermal drift of resistance can shift calibration; metal film or manganin resistors help stabilize performance. Full-scale deflection current ( I g ) The maximum current the galvanometer coil can carry for its pointer to reach full-scale safely. Radial field A magnetic field directed radially with respect to the coil’s axis so that B is nearly perpendicular to the coil over its motion, ensuring linearity. Direction practice: In a horizontal magnetic field pointing east, a vertical wire carrying upward current experiences a force toward north (using the right-hand rule). Reverse the current or the field and the force reverses. This physical intuition checks your algebra: if your numeric answer is positive but the geometry suggests the opposite direction, re-evaluate the cross product direction. A 12 cm wire carries 4.0 A and is placed perpendicular to a 0.30 T magnetic field. Find the acceleration of the 10 g wire segment if it is free to slide horizontally on frictionless guides. F = B I L 90 = B I L. Then a = F/m. Kinematics-meets-magnetism application L = 0.12 , m I = 4.0 , A B = 0.30 , T m = 10 , g = 0.010 , kg = 90 Acceleration a medium This dynamic example shows that magnetic forces can produce linear acceleration if the conductor is free to move. In many meter designs, however, mechanical constraints transform the same magnetic interaction into rotation rather than translation, which is why torque formulas dominate instrument analysis. Practice translating verbal descriptions to the correct mathematical angles. Words like “plane of the coil” and “normal to the plane” map to vectors that are 90° apart. Build a habit: draw a quick sketch, mark the area vector, and then read off the correct θ for sinθ. This saves many marks on quick NEET calculations. Do not confuse figure of merit k with the coil resistance G. k links current and deflection (I = k θ), while G links current and voltage (V = I G). Using G where k is needed leads to dimensional nonsense. neet-alert Summary points: A straight conductor in a uniform magnetic field feels a force F = B I L sinθ perpendicular to both I and B. A closed loop has zero net force in uniform B but experiences torque τ = n B I A sinθ or τ = μ × B, seeking alignment of μ with B. A moving-coil galvanometer balances magnetic torque and spring torque (n B I A = k θ), yielding a linear scale for small angles. Converting to an ammeter requires a low shunt R s = (G I g )/(I − I g ); converting to a voltmeter requires a high series resistor R series = V/ I g − G. Watch angles and units to avoid traps. Key terms recap Net force on a current-carrying wire in a magnetic field: F = B I L sinθ. Magnetic force on conductor Torque on loop Twisting effect on a current loop: τ = n B I A sinθ or τ = μ × B. dipole moment For a loop, μ = n I A; direction by right-hand rule. Magnetic dipole moment Galvanometer Instrument where magnetic torque and spring torque balance to measure small currents. Figure of merit k = I/θ (current per unit deflection). FOM Low parallel resistor that bypasses current to extend ammeter range. R s Shunt multiplier R series High series resistor used to extend voltmeter range. Series multiplier S i S i = θ/I = n B A / k. Current sensitivity Voltage sensitivity S v S v = θ/V = (n B A)/(k G).