To determine the total heat produced in the necklace, we can use the principles of electromagnetic induction. According to Faraday's Law, a changing magnetic field through a loop induces an electromotive force (emf) in the circuit. This emf leads to a current through the resistive material, generating heat.
The magnetic flux (\(\Phi\)) through the necklace enclosed area can be expressed as:
The formula for magnetic flux is:
\(\Phi = B \times A\)
Initially, before the power failure:
\(\Phi_{\text{initial}} = 2 \, \text{T} \times 0.01 \, \text{m}^2 = 0.02 \, \text{Wb}\)
After the power failure, the field drops to 1T:
\(\Phi_{\text{final}} = 1 \, \text{T} \times 0.01 \, \text{m}^2 = 0.01 \, \text{Wb}\)
The change in magnetic flux is:
\(\Delta \Phi = \Phi_{\text{final}} - \Phi_{\text{initial}} = 0.01 \, \text{Wb} - 0.02 \, \text{Wb} = -0.01 \, \text{Wb}\)
Since the time \(\Delta t = 10^{-3} \, s\), the induced emf is:
\(E = -\frac{{\Delta \Phi}}{{\Delta t}} = -\frac{{-0.01 \, \text{Wb}}}{{10^{-3} \, s}} = 10 \, \text{V}\)
Where \(E\) is the emf, \(R\) is the resistance, and \(t\) is the time duration.
Substitute values:
\(H = \frac{{(10 \, \text{V})^2 \times 10^{-3} \, \text{s}}}{{0.01 \, \Omega}} = \frac{{100 \, \text{V}^2 \times 10^{-3} \, \text{s}}}{{0.01 \, \Omega}}\)
Simplifying:
\(H = \frac{{100 \times 10^{-3}}}{{0.01}} = 10 \, \text{J}\)
The total heat produced in her necklace is 10 J.