A hollow metal cylinder has inner radius a, outer radius b, length L, and conductivity σ. The current I is radially outward from the inner surface to the outer surface. Evaluate the electric field strength at the inner and outer surfaces of an iron cylinder if a=1.0 cm, b=2.5 cm, L=10 cm, and I=25 A.

Verified step by step guidance

1. Understand the problem: The current flows radially outward through the hollow cylindrical conductor. The electric field strength at the inner and outer surfaces can be determined using Ohm's law in its differential form, which relates the current density, electric field, and conductivity.
2. Step 1: Write down the formula for current density (J). The current density is given by: $J=I/A$, where $I$ is the current and $A$ is the cross-sectional area through which the current flows. For a hollow cylinder, the cross-sectional area at a radius r is: $A=2\pi rL$, where $L$ is the length of the cylinder.
3. Step 2: Relate the electric field to the current density using Ohm's law. Ohm's law in its differential form is: $E=J/\sigma$, where $E$ is the electric field strength, $J$ is the current density, and $\sigma$ is the conductivity of the material.
4. Step 3: Calculate the electric field at the inner surface (r = a). Substitute $r=a$ into the expressions for $A$ and $J$, then use Ohm's law to find $E$ at the inner surface.
5. Step 4: Calculate the electric field at the outer surface (r = b). Similarly, substitute $r=b$ into the expressions for $A$ and $J$, then use Ohm's law to find $E$ at the outer surface.

Key Concepts

Ohm's Law

Ohm's Law states that the current (I) flowing through a conductor between two points is directly proportional to the voltage (V) across the two points and inversely proportional to the resistance (R) of the conductor. This relationship can be expressed as V = IR. In the context of the hollow cylinder, understanding Ohm's Law is essential for calculating the electric field and voltage drop across the material.

Electric Field in Conductors

The electric field (E) within a conductor is related to the current density (J) and conductivity (σ) by the equation J = σE. In a cylindrical conductor, the electric field can be determined by analyzing the radial symmetry and the distribution of current. This concept is crucial for evaluating the electric field strength at the inner and outer surfaces of the hollow cylinder.

Current Density

Current density (J) is defined as the amount of electric current flowing per unit area of a cross-section of the conductor. For a hollow cylinder, the current density can be expressed as J = I/A, where A is the cross-sectional area through which the current flows. Understanding current density is important for calculating the electric field and analyzing how the current is distributed across the cylinder's surfaces.