When the current in a toroidal solenoid is changing at a rate of 0.0260 A/s, the magnitude of the induced emf is 12.6 mV. When the current equals 1.40 A, the average flux through each turn of the solenoid is 0.00285 Wb. How many turns does the solenoid have?

Verified step by step guidance

1. Identify the given values: rate of change of current ($\frac{dI}{dt}$) is 0.0260 A/s, induced emf ($\mathcal{E}$) is 12.6 mV or 0.0126 V, current (I) is 1.40 A, and average flux per turn ($\mathrm{\Phi }$) is 0.00285 Wb.
2. Use Faraday's Law of Electromagnetic Induction which states $\mathcal{E}=N\frac{d\mathrm{\Phi }}{dt}$, where N is the number of turns in the solenoid and $\frac{d\mathrm{\Phi }}{dt}$ is the rate of change of magnetic flux.
3. Calculate the total change in flux ($\frac{d\mathrm{\Phi }}{dt}$) using the relationship between flux and current. Since $\mathrm{\Phi }=N\cdot \varphi$ where $\varphi$ is the flux per turn, and the current is changing, use $\frac{d\mathrm{\Phi }}{dt}=N\cdot \frac{d\varphi }{dt}$.
4. Substitute the expression for $\frac{d\mathrm{\Phi }}{dt}$ in terms of $\frac{dI}{dt}$ and $\varphi$ into Faraday's Law equation. Solve for N by rearranging the equation to isolate N on one side.
5. Plug in the values for $\mathcal{E}$, $\frac{dI}{dt}$, and $\varphi$ to calculate the number of turns, N.

Key Concepts

Faraday's Law of Electromagnetic Induction

Faraday's Law states that the induced electromotive force (emf) in a closed loop is directly proportional to the rate of change of magnetic flux through the loop. This principle is fundamental in understanding how changing currents can generate voltage in circuits, particularly in inductive components like solenoids.

Magnetic Flux

Magnetic flux is defined as the product of the magnetic field strength and the area through which the field lines pass, taking into account the angle between the field lines and the normal to the surface. It is measured in webers (Wb) and is crucial for calculating the induced emf in a solenoid when the current changes.

Toroidal Solenoid

A toroidal solenoid is a coil of wire shaped into a doughnut form, which creates a magnetic field when an electric current passes through it. The geometry of the toroid allows for a uniform magnetic field inside the coil, and its design is significant in applications where space and field uniformity are important.