A conducting disk with radius \(a,\) thickness \(h,\) and resistivity \(\rho\) is inside a solenoid of circular cross section, its axis coinciding with the solenoid axis. The magnetic field in the solenoid is given by \(B=b t,\) where \(b\) is a constant. Find expressions for (a) the current density in the disk as a function of the distance \(r\) from the disk center and (b) the power dissipation in the entire disk. (Hint: Consider the disk as consisting of infinitesimal conducting loops.)

Faraday's Law of Electromagnetic Induction is a fundamental principle stating that a change in magnetic flux through a conducting loop induces an electromotive force (EMF) in the conductor. The inducing magnetic field's rate of change is key to determining the size of the induced EMF.

Our exercise involves a conducting disk inside a solenoid, where the magnetic field varies with time. According to Faraday's Law, the magnetic flux \( \Phi \) through the disk changes as \( B \) changes, where \( B = bt \) indicates the magnetic field grows linearly with time and \( b \) is a constant. For any small slice of the disk at a distance \( r \) from the center, the flux is described as \( \Phi = B \cdot \pi r^2 \).

The negative derivative of magnetic flux with respect to time gives us the EMF. This leads to an electric field \( E \) induced in the disk, crucial for generating currents and leading to effects such as current density and power dissipation, as explored in the steps following this principle.

Current density, denoted by \( J \), is a measure of the electric current per unit area flowing through a material at any given point. It is a vector quantity with both magnitude and direction and is expressed by the formula \( J = \frac{I}{A} \), where \( I \) is the electric current and \( A \) is the cross-sectional area.

In the context of our problem, after the electric field \( E = \frac{br}{2} \) is calculated using Faraday's Law, Ohm's Law (\( J = \frac{E}{\rho} \) where \( \rho \) is the resistivity) is applied to find the current density as a function of \( r \) within the disk. With the given relationship \( E = \frac{br}{2} \) and material's resistivity \( \rho \) as constants, the current density \( J \) inside the disk varies linearly with the distance \( r \) from its center, leading to \( J = \frac{br}{2\rho} \).

Understanding this relationship helps in analyzing how electric charge distribution changes across the disk—a concept that has real-world implications for designing effective electromagnetic devices.

Power dissipation in electrical systems refers to the process by which an electrical device converts electric power into another form of energy, often heat, as a result of resistance. The power lost can be described using the formula \( P = I^2R \) or, equivalently, \( P = J^2 \rho A \) where \( P \) is power, \( I \) is current, \( R \) is resistance, \( J \) is current density, and \( A \) is the area.

In the exercise, we calculate the power dissipation for an infinitesimally small slice of the conducting disk. By substituting \( J \) and the differential resistance \( dR \) into the power dissipation formula, we obtain the expression \( dP = \frac{b^2 r dr}{4 \pi h} \) for an infinitesimal slice. To find the total power dissipation across the entire disk, we integrate this expression from the center of the disk (\( r = 0 \)) to the edge (\( r = a \)). This results in the formula \( P_{total} = \frac{b^2 a^3}{12 \pi h} \) which embodies the collective heat energy lost due to the resistance encountered by current in the disk.

This concept is widely applicable in electronics and electrical engineering, as understanding how materials convert electrical energy to heat aids in managing system efficiency and longevity.