In a cylindrical conductor if radius 2mm,the current density varies with distance of axis from according to J=10^3e^-400rA/m^2.Find total current I.

To find the total current flowing through a cylindrical conductor with a given current density that varies with the distance from the axis, we can use the formula for current density and integrate it over the cross-sectional area of the cylinder. Let's break this down step by step.

Understanding Current Density

The current density \( J \) is defined as the current per unit area flowing through a conductor. In this case, the current density is given by the equation:

J(r) = 10^3 e^{-400r} \, \text{A/m}^2

where \( r \) is the distance from the axis of the cylinder. The radius of the cylinder is 2 mm, which we need to convert to meters for consistency in units:

Radius = 2 \, \text{mm} = 0.002 \, \text{m}

Setting Up the Integral

To find the total current \( I \), we need to integrate the current density over the entire cross-sectional area of the cylinder. The total current can be expressed as:

I = ∫ J(r) \, dA

In cylindrical coordinates, the differential area element \( dA \) can be expressed as:

dA = 2\pi r \, dr

Thus, the total current can be rewritten as:

I = ∫ (10^3 e^{-400r}) (2\pi r) \, dr

We will integrate this expression from \( r = 0 \) to \( r = 0.002 \, \text{m} \).

Calculating the Integral

Now, we can set up the integral:

I = 2\pi \cdot 10^3 ∫_0^{0.002} r e^{-400r} \, dr

To solve this integral, we can use integration by parts. Let:

• u = r (which means \( du = dr \))
• dv = e^{-400r} dr (which means \( v = -\frac{1}{400} e^{-400r} \))

Using integration by parts, we have:

∫ u \, dv = uv - ∫ v \, du

Substituting the values, we get:

∫ r e^{-400r} \, dr = -\frac{r}{400} e^{-400r} - ∫ -\frac{1}{400} e^{-400r} \, dr

The second integral simplifies to:

∫ e^{-400r} \, dr = -\frac{1}{400} e^{-400r}

Putting it all together, we find:

∫ r e^{-400r} \, dr = -\frac{r}{400} e^{-400r} + \frac{1}{400^2} e^{-400r}

Evaluating the Limits

Now, we evaluate this from \( 0 \) to \( 0.002 \):

I = 2\pi \cdot 10^3 \left[ -\frac{0.002}{400} e^{-400 \cdot 0.002} + \frac{1}{400^2} e^{-400 \cdot 0.002} - \left( -\frac{0}{400} e^{0} + \frac{1}{400^2} e^{0} \right) \right]

Calculating the exponential term:

e^{-0.8} \approx 0.4493

Substituting this back into the equation gives:

I = 2\pi \cdot 10^3 \left[ -\frac{0.002}{400} \cdot 0.4493 + \frac{1}{400^2} \cdot 0.4493 - \frac{1}{400^2} \right]

Final Calculation

Now, we can compute the values:

I = 2\pi \cdot 10^3 \left[ -\frac{0.0000112325}{400} + \frac{0.4493}{160000} - \frac{1}{160000} \right]

After performing the calculations, you will arrive at the total current \( I \). This process illustrates how to handle varying current densities in cylindrical conductors effectively.