Generate an emf by changing loop area

Change area while the field stays steady

The previous coil investigation changed the field through a stationary area. Here the magnetic field is steady, but a conducting rod moves along two rails and changes the enclosed area. The left resistor, upper rail, rod and lower rail form one closed conducting loop. The rails extend beyond the rod; those open extensions are not extra closed loops.

The rod spans ℓ = 0.50 m between the rails. Its initial position is x = 2 m from the left connection. An external driver maintains velocity v for 0.50 s. The circuit’s total resistance is 2 Ω. The diagram shows the final configuration and a faint initial-position marker; it does not animate an accelerating object. Horizontal positions use a common scale, while the vertical dimension is enlarged. Field symbols represent a uniform field throughout the modeled region, not a field confined by the moving rod.

At v = +1 m/s and Bz = +1 T, predict whether flux increases and which way current flows. Positive z points toward you; positive loop circulation is counterclockwise, upward along the rod.

Compute the area and signed flux

Position follows x(t) = x₀ + vt, with x₀ = 2 m. At 0.50 s and +1 m/s, x = 2.50 m. The enclosed area A = ℓx grows from 1.00 to 1.25 m². These are deliberately clear teaching dimensions, not an apparatus construction plan.

With normal +z and a uniform signed field Bz, Φ = Bz ℓx. At +1 T, flux grows from +1.00 to +1.25 Wb. The flux slope is ΔΦ/Δt = 0.25/0.50 = +0.50 Wb/s. Faraday’s law gives a signed loop emf of −0.50 V. The negative sign means clockwise circulation in the chosen view.

Because Bz and ℓ are constant during the run, the rate of area change is ℓv and emf = −Bz ℓv. The units are tesla × metre × metre/second, equivalent to weber/second or volt. This signed expression is specific to the chosen loop normal and rod orientation. The familiar magnitude |B|ℓ|v| omits circulation information and assumes mutually perpendicular rod length, velocity and field.

Check the direction using moving charges

For positive charge moving with the rod, the magnetic force direction is q(v × B). With velocity rightward (+x) and field toward you (+z), the magnetic contribution points down the rod (−y). This agrees with clockwise current in the closed loop. Conventional current is the positive-charge direction; do not reverse it a second time because conduction electrons have negative charge.

The motion-induced emf is associated with the moving conductor. It differs from the previous stationary loop driven by a time-changing field, but both agree with the changing-flux calculation for their respective moving or stationary boundaries. Using only field change and ignoring area change would incorrectly predict zero emf here.

Test reversals separately

Capture the default case, then reverse only Bz to −1 T. The loop area still grows, but its signed flux becomes more negative. Emf becomes +0.50 V and current becomes counterclockwise. Reverse only velocity from the default instead: the rod moves left, enclosed area shrinks, and the emf again becomes +0.50 V. Reversing both field and velocity restores the original emf sign.

At zero velocity, flux is nonzero if the field is nonzero, but it stays constant and emf is zero. At zero field, this model has zero magnetic flux, emf and current at any available velocity. The rod can still move because its velocity is prescribed externally.

The interval and controls keep x between 1 and 3 m, so the rod never crosses the left connection. There is no collision, endpoint switch or changing contact modeled. Changing a slider selects a different constant-velocity scenario starting from the same initial position, not a mid-run intervention.

Distinguish emf from a closed-circuit current

With total resistance R = 2 Ω, the resistance-only approximation gives I = emf/R. At the default −0.50 V, current is −0.25 A, clockwise. Negligible inductance and rapid charge redistribution are assumptions; switching transients are outside this model.

A moving isolated rod can develop an end-to-end voltage through charge separation without supporting a sustained closed-loop current. For the default rightward motion and outward field, its lower end would be positive relative to its upper end in the ideal open-circuit limit. That is a different circuit condition from the browser’s closed loop, not an extra current path. Opening the loop is not a control here, and voltage alone does not establish delivered electrical power.

Transfer to another rod

A 0.30 m rod moves right at 0.80 m/s in a 0.50 T field pointing into the page. It closes a loop through 4 Ω with the same geometry and +z normal. Find the signed emf and current. Predict the magnitude if speed doubles.

Check the motional-emf transfer

Bz = −0.50 T, so emf = −(−0.50) × 0.30 × 0.80 = +0.12 V. Current is +0.12/4 = +0.030 A, counterclockwise in this view. Doubling speed doubles emf magnitude to 0.24 V and current magnitude to 0.060 A under the same resistance-only assumptions. Rod length and resistance differ from the browser constants.

Use captured flux endpoints to calculate an emf independently of the voltage display. State the loop reference, whether area or field changed, and why a closed conducting path is required for the displayed current.

Reference for the moving-boundary relation: OpenStax: motional emf. The signed scenarios and transfer exercise are original.