A solenoid of radius R=5.00 cm is made of a long piece of...

A solenoid of radius $R=5.00 \mathrm{~cm}$ is made of a long piece of wire of radius $r=2.00 \mathrm{~mm}$, length $\ell=10.0 \mathrm{~m}$ $(\ell \gg R)$ and resistivity $\rho=1.70 \times 10^{-8} \Omega \cdot \mathrm{m} .$ Find the magnetic field at the center of the solenoid if the wire is connected to a battery having an emf $\mathcal{E}=20.0 \mathrm{~V}$.

A solenoid of radius $R=5.00 \mathrm{~cm}$ is made of a long piece of wire of radius $r=2.00 \mathrm{~mm}$, length $\ell=10.0 \mathrm{~m}$ $(\ell \gg R)$ and resistivity $\rho=1.70 \times 10^{-8} \Omega \cdot \mathrm{m} .$ Find the
magnetic field at the center of the solenoid if the wire is connected to a battery having an emf $\mathcal{E}=20.0 \mathrm{~V}$.

Key Concepts

Ohm's Law

Ohm's law relates the current flowing through a conductor to the applied voltage and the resistance offered by the conductor. It is a fundamental concept in circuit analysis, expressed as I = V/R, where I is the current, V is the voltage, and R is the resistance. This relationship allows us to calculate the current supplied by a battery once the total resistance of the circuit is known.

Electrical Resistance and Resistivity

Electrical resistance quantifies how much a material opposes the flow of electric current, and it depends on both the material's intrinsic resistivity and its geometric dimensions. The resistance of a wire is calculated using the formula R = ?L/A, where ? is the resistivity, L is the length of the conductor, and A is its cross-sectional area. Understanding this concept is crucial for determining how much current will flow when a voltage is applied across a given piece of wire.

Solenoid Geometry and Number of Turns

The geometry of a solenoid defines its magnetic properties, particularly through the number of turns per unit length. When a long wire is wound into a tight coil of a given radius, each turn has a circumferential length related to the solenoid’s radius. The total number of turns can be determined by dividing the total length of the wire by the circumference of one turn. This ratio is key in quantifying the density of the coils, which directly affects the strength of the magnetic field inside the solenoid.

Magnetic Field of a Solenoid

The magnetic field inside an ideal solenoid is uniform and is given by the expression B = ??nI, where ?? is the permeability of free space, n is the number of turns per unit length, and I is the current flowing through the wire. This relationship explains how both the current and the coil's density work together to generate a magnetic field at the center of the solenoid, making it a central concept in electromagnetism and magnetic field design.

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