A long, thin solenoid has 400 turns per meter and radius 1.10 cm. The current in the solenoid is increasing at a uniform rate di/dt. The induced electric field at a point near the center of the solenoid and 3.50 cm from its axis is 8.00 × 10^-6 V/m. Calculate di/dt.

Verified step by step guidance

1. Start by understanding the concept of electromagnetic induction in solenoids. When the current in a solenoid changes, it induces an electric field around it. This is described by Faraday's law of induction.
2. Use the formula for the induced electric field inside a solenoid: $E=\frac{r}{R}\frac{d}{i}dt$, where $E$ is the induced electric field, $r$ is the distance from the axis, $R$ is the radius of the solenoid, and $\frac{d}{i}dt$ is the rate of change of current.
3. Substitute the given values into the formula: $8.00\times {10}^{-}$ for $E$, $3.50$ cm for $r$, and $1.10$ cm for $R$. Convert these distances to meters.
4. Rearrange the formula to solve for $\frac{d}{i}dt$: $\frac{d}{i}dt$ = ERr.
5. Calculate the value of $\frac{d}{i}dt$ using the rearranged formula and the substituted values.

Key Concepts

Faraday's Law of Induction

Faraday's Law states that a changing magnetic field within a closed loop induces an electromotive force (EMF) in the loop. In the context of a solenoid, the changing current alters the magnetic field inside, inducing an electric field around the solenoid. This principle is crucial for understanding how the rate of change of current (di/dt) affects the induced electric field.

Magnetic Field Inside a Solenoid

The magnetic field inside a long, thin solenoid is uniform and given by B = μ₀nI, where μ₀ is the permeability of free space, n is the number of turns per unit length, and I is the current. This concept helps in understanding how the solenoid's properties and current contribute to the magnetic field, which is essential for calculating the induced electric field.

Induced Electric Field

The induced electric field around a solenoid is related to the rate of change of the magnetic field inside it. According to Faraday's Law, the magnitude of the induced electric field at a distance from the solenoid's axis can be calculated using the formula E = (r/2) * (dB/dt), where r is the radial distance from the axis. This concept is key to determining the relationship between di/dt and the given electric field.